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

Linear and second order nonlinear gratings induced by two photon absorption in thermally poled glasses. S. Montant, E. Freysz Centre de Physique Moleculaire Optique et Hertzienne Universite Bordeaux 1 354, cours de la liberation 33405 T ALENCE CED EX, France Tel: (33) 5 56 84 62 06, Fax: (33) 5 56 84 69 70, e-mail: montant@cribxl.u-bordeaux.fr Non phase matched second harmonic generation (SHG) can be produced in thermally poled fused silica [1]. The second order susceptibility z i <;J induced over -10 µm beneath the anodic surface is due to coupling of the third order glass susceptibility x <3l with a built in electric field Em1 ( zi:J -x <3l E;n1). To achieve quasi-phase matching (QPM) conditions, different techniques have been proposed and a 20% SHG efficiency has been demonstrated in QPM thermally poled fibers [2]. Here using amplified femtosecond pulses in thermally poled glasses, we propose a new method which results within a couple of minutes in the formation of both thick linear and non linear QPM thin gratings. Our results also show that thermally poled glasses behave as erasable optical memories. The experimental set-up, presented in Figure 1, uses femtosecond non amplified (76 MHz C.W mode-locked, 120 fs, 10 nJ) and amplified ( 1 kHz, 0.5 mJ) Ti: Sapphire laser pulses centered at 810 nm as probe and pump beams, respectively. The collinear pump and probe pulses, linearly and orthogonally polarized respectively, were sent through a phase grating ( 333 grooves/cm). After the grating the two equally balanced and focussed ( w= 160 µm) pump beams were overlapped and produced a periodic intensity pattern ( l(r)=l0(l+cos(2n.r/A) where A=3.3 µm) in the thermally poled 30% Nb205 doped borophosphate glass sample. The samples were poled at 300°C for about 1 hour applying a de bias of 3 kV 1 across the 1 mm thick glass. They were tested using weak probe beams (Epr-0.5 nJ). With a single probe beam going through the sample, steady SHG was recorded along the beam direction, whereas with the two synchronized and overlapping probe beams, SHG was detected in both the probe beams and in the non collinear SHG direction. Then, the pump beam modulated by a mechanical chopper was sent through the sample. The temporal evolution of the probe SHG recorded when the pump beam was blocked, is presented in Figure 2. These figures demonstrate the pump beams erase the zi � .( Fig. 2 a-b ). The temporal decay of the SHG signal S(r,2ro,t) well agree with a one dimension spatial extension of the decay law , we recently proposed [3] (Fig. 2a-b full-line). Indeed, according to the two photon-erasure process one can consider that the time dependent SHG signal evolve as:

S(r,2m,t = O) = ( z � ! / < ; �, t = 0) ]2 = 1 + fi (m)/2 (r,m} (I) S(r,2m,t) z ind (r,t)
where l(r,ro) describe the spatial profil of the erasure intensity pattern, �(ro) is a coefficient which depends on the two photon absorption of the used glass and zi � (r,t) describe the spatial and temporal evolution of the induced second order nonlinearity during the erasure process. Note that at the beginning of the erasure process, we verified S(r,2w,t=O) oc (Z i � (r,t=0))2 was almost constant over the whole thermally poled region. We also tried to evidence the formation of a thin non-linear grating zi � (r). After the erasure, if one shines this sam ple with a single probe laser beam one would expect to record SHG along different directions. Indeed, one can easily show that the far field second harmonic intensity pattern generated inside the thin poled region by the spatially modulated zi � (r,t) is proportional to the square modulus of the spatial Fourier transform of zi � (r,t). In figure 2c we have reported the evolution of the intensity ratio of harmonic generation along the first and second order of diffr action of the induced nonlinear grating.

One can note that while the trend is well described the experimental ratio is down shifted compared to the theoretically expected ratio. We believe this clearly indicate the limitation of the one spatial dimension model we proposed. --------------------------------------------I -theory Fig. 2 : a-b:Theoretical ( full line) and experimental evolution of the maximum of the intensity autocorrelation (0) and of the first diff raction order (Li) of the probe beams during second order nonlinear grating formation. c: Theoretical (full line) and experimental (dotted line) evolution of the ratio between the second harmonic intensity recorded in the first and second order Bragg direction. To record the signal the pump is blocked and a single probe beam is sent through the sample.

--•--Ezperiment
The diffusion of carriers created by either one or two-photon absorption of the pump beams should reduce fqn1 and may explain the SH signal relaxation [3] or the increase of the %� in UV-excited poled fibers. This might be associated with the formation of photo-refractive linear index grating in the whole sample volume. To evidence this, an experiment was performed using an He-Ne laser beam entering the glass sam ple at a Bragg diffraction angle. The temporal evolution of the diffracted He-Ne beam is 1 presented in Figure 3. When the pump beam was on, its intensity rapidly increased and saturated. Its intensity decreases slowly when the pump beam was off. Under this experimental condition we only record a light di ffraction along the Bragg direction. We have not been able to observed any increase of the sample absorption after the erasure process. These results clearly indicate we are recording the formation and relaxation of a thick linear index grating. The power dependence of the grating reflectivity( i.e.11-r' Jl'l• Fig. 2b) [4] and the formation (t-1"2pu Fig. 2c) (5) time agree with a two-photon photo-refractive phenomenon. We evaluated the maximum index change to be: .&1-5*10"5• Similar thick index grating was also recorded in a non-poled sample.

u �(a) Finally, our results also show that thermally poled glasses can be used as erasable optical memories: one can consider that the spatially limited optical eraser of the induced second order non linearity allows to store a bit of information that can be retrieved looking for the absence of SHG at the considered point. In fact, we have shown that poled glasses have less binary level of erasure and have photographic film like behavior.

Epr

Figure 1 :

 1 Figure 1: Experimental set-up

o

  Experimen

Figure 3 .

 3 Figure 3. a: Formation and relaxation of linear index grating. b: Evolution of steady state grating reflectivity 17 as a function of pump energy. c: Evolution of grating formation time -r as a function of pump energy


From Boccaccio to Petrarch, from medical treatises to historical chronicles, from astrological explanations to theological speeches, the medieval plague inspired people to note their experiences. In the early fifteenth century, the University of Vienna was confronted with several outbreaks of plague of varying intensity, from the very violent, with many victims among the students and masters, to more calm episodes. The great historian of the university Joseph Aschbach listed visits of the plague in "1400, 1419, 1425, 1428, 1442, 1444, 1452 u. A." 1 This paper will present testimony to another outbreak recorded at the University of Vienna in 1436, offering some interesting details about the impact of the medical situation on the academic calendar and on the careers of some scholars. In fact, the incident and the testimony have been mentioned on occasion in catalogues and repertories, but only fragmentarily. [START_REF] Rapf | Magister Iohannes Straidler collegiatus fuit admissus ad legendum Sentencias, sed nichil[END_REF] The appendix to this paper offers the first full transcription, all 37 lines.

At first glance, nothing in Wien, Österreichische Nationalbibliothek, cod. 4497, indicates any obvious connection to the plague, since the manuscript contains a copy of the Sentences commentary of Johannes Grössel de Tittmoning (Johannes Groessel de Tytmaning), 3 who was assigned to begin reading the Sentences in the Faculty of Theology during the 1435-1436 academic year, although he would not obtain his licence in theology until 1448. 4 At the end of Book II in Wien ÖNB 4497, one finds a strikingly long colophon, an emotional account of a visit of the plague that affected the teaching schedule at the University of Vienna.

Johannes Grössel was a prominent figure at the University of Vienna, serving four times as dean of the Faculty of Arts (1432, 1438, 1444, and 1447), eight times as dean of the Faculty of Theology (1451, 1453, 1456, 1458, 1462, 1463, 1464, and 1465), and four times as rector of the university (1436, 1443, 1455, and 1465). 5 This chronology shows that Johannes Grössel was an authoritative witness to the events at the university during various epidemics, since he was occupied with important administrative duties. In 1436, when in the middle of his Sentences commentary he reports what happened in the Faculty of Theology, he was not merely a young * The research for this paper was done within the framework of the project ERC-CoG-DEBATE n° 771589, https://debate-erc.com/.

1 J. Aschbach, Geschichte der Wiener Universität im ersten Jahrhundert ihres Bestehens, Wien 1865, p. 334. [START_REF] Rapf | Magister Iohannes Straidler collegiatus fuit admissus ad legendum Sentencias, sed nichil[END_REF] For example, in his description of the manuscript Michael Dennis refers to the note on the plague but rewrites the contents and reports just a portion of the information, while Friedrich Stegmüller reproduces a couple of lines, as does Aschbach: M. Denis, Codices manuscripti theologici bibliothecae palatinae Vindobonensis Latini aliarumque occidentis linguarum, Wien 1800, vol. II, pars scholar, but an experienced administrator who became rector in the winter semester of 1436-1437 just after the outbreak, perhaps because so many of his senior colleagues fell victim to the plague.

On folio 266r of Wien, ÖNB, 4497, the lament begins by informing us that the author of the note started to write and to lecture on Book III of the Sentences on the Monday immediately following the Octave of Easter in 1436, i.e., 16 April, and finished one day before the feast of St. Nicholas, namely on 5 December of the same year. The reason it took such an extraordinarily long time to complete his lecture series is because the epidemic paralyzed the university with especially dramatic intensity in August, when, before the feast of the Assumption of the Blessed Virgin, 70 members of the University of Vienna were buried in one day. Johannes lists a few of the most notable of those who perished around that time, starting from the highest position, his professors, and gradually descending to the lowest, his socii, or the fellows with whom he debated or was supposed to debate before the assembled faculty in the principia sessions before each book of the Sentences. Leaving aside unnamed virtuous masters, bachelors, and students, Grössel lists the following individually:

Master Petrus Reicher de Pirchenwart, professor of the Sacred Page, senior regent in the Collegium Ducale and lector in theology.

Master Conradus de Herbst of the Order of Preachers, likewise a doctor, although a new one and not of one year, buried in the house of the Preachers of Vienna.

Master Urbanus de Melk, canon of the Church of St Stephan and doctor in theology.

Master Johannes Stredler de Langhüta, formed bachelor in theology and collegiatus in the Collegium Ducale, who died in his college.

Two unnamed masters of the Collegium Ducale who died a few days later.

Facing this dramatic situation, the university assembled and decided to suspend classes and all scholastic acts, sermons, and administrative meetings from just before 15 August, when it seems it was the peak of the outbreak, until the feast of St. Coloman, on 13 October, unless it were deemed expedient to extend the suspension. After responding in the annual aula disputations for advanced theology students in the Collegium Ducale, 6 with the university entering a full vacation period, Johannes Grössel de Tittmoning himself immediately returned to his native land in Bavaria on the current border with Austria, departing on Sunday 12 August, before the feast of the Assumption. Grössel remained in seclusion at home for 13 weeks and one day, he says, going back to Vienna on the Tuesday after the feast of St. Elizabeth. Presumably this was Elizabeth of Hungary, in which case he returned on 20 November, although this entails an absence of 14 weeks and one day. Arriving in Vienna, he found only four fellows of the Collegium Ducale who were still there, saying, "with them, I was the fifth." Indeed, Grössel remarks that after the suspension of classes and his own departure only one of 6 The precise place inside the Collegium Ducale for debates with socii or lectures was in stuba collegii or in aula. For the architectural structure of this college in Vienna see K. Mühlberger, "The Old University Quarter. The Medieval University," in J. Rüdiger, D. Schweizer, translated by N.K. Müngersdorff (eds.), Sites of Knowledge. The University of Vienna and Its Buildings: A History 1365 -2015, Wien-Köln-Weimar, 2015, pp. 13-42. For a recent discussion of the aula and questiones in aula see D. Coman, "Grace Meets Free Will Ruling in a Regal Government: Master Michael Suchenschatz on Grace and Free Will," in M. Brînzei (ed.), The Rise of an Academic Elite: Deans, Masters, and Scribes at the University of Vienna before 1400 (Studia Sententiarum, 6), Turnhout, Brepols, 2022, forthcoming.

the masters presided in the Collegium Ducale until the feast of St. Coloman on 13 October, the normal start of classes.

Normal activities only resumed belatedly on the feast of St. Catherine on Sunday, 25 November, when the doctors and masters assembled, the new administrators were chosen, and the subjects of the lectures were assigned to the instructors in all faculties, all much later than was normal. This gap is also visible in the Acta Facultatis Theologie, where the records break off with the last note from 6 August 1436 and the next entry is only made on 19 December 1436 when two new fellows, Wolffgangus de Lapide and Paulus de Melk, were admitted to the Faculty of Theology. Nothing about the suspension of classes or the loss of faculty members is mentioned. [START_REF]Akten der Theologischen Fakultät der Universität Wien[END_REF] In contrast to the silence of the Acta, the emotional testimony of Johannes Grössel offers a realistic inside view on what happened at the University of Vienna in late 1436. Let us add some details to his account.

The first major figure Johannes Grössel mentions is Petrus de Pirchenwart, a famous theologian and student of Petrus de Pulkau. [START_REF]Stanislaus of Znojmo and the Arrival of Wyclif's Remanence Theory at the University of Vienna[END_REF] He was also one of the first masters in Vienna to discuss the theory of transsubstantiation of Stanislaus of Znojma, the inspirational master of Jan Huss, and Pirchenwart also wrote a treatise against the Hussites. 9 In the Acta Facultatis Theologie of the University of Vienna, the last note on Pirchenwart is from 1 March 1436, when he was assigned to deliver the university sermon on Pentecost, which fell on 27 May that year. 10 As Grössel notes, Pirchenwart's activity ended tragically around the feast of the Assumption.

For Urban of Melk we have a detailed intellectual portrait by Martin Wagendorfer. 11 Urban was an important figure in the Faculty of Arts and only after two decades of teaching in Arts did he enrol as a student in the Faculty of Theology. He was licensed in 1433 and had his aula disputation on 2 May 1436, which is why Grössel calls him a doctor of theology. It is possible that his illness was long enough to allow him time to write his last will, since he left seven beloved books to the library of the Schottenkloster. [START_REF] For | Ein Bücherstestat an das Wiener Schottenstift[END_REF] His example illustrates the circulation of books among the members of the university, revealing how books were inherited, purchased, or donated by scholars, [START_REF]containing Augustine's De Civitate Dei, today under the selfmark Schottenstifts, cod. 20[END_REF] especially during visits of the plague. Wegendorfer also comments on the episode of Urban's death, since he found a detailed note on this event in manuscript Schottenstift 363, [START_REF]This note is also mentioned in Gottlieb's history of Austrian libraries, but it can be read in its extension exprimens, verum Messiam venisse, contra[END_REF] which contains a report similar to that of Grössel. In the note in Schottenstift 363, we are informed that the epidemic effectively began on the feast of St. Margaret on 10 June and lasted until 15 August 1436. During this period, 11,000 persons died in Vienna, so it is easy to believe that, in only one day, the university buried 70 scholars. It is here that we have the precise indication of Urban's death, which occurred on 9 August, one day before the feast of St. Laurence (in vigilia sancti Laurentii obit magister Urbanus de Melico). 15 If the plague arrived on 10 June 1436, it came just after the promotion of the Dominican Conrad of Herbest, who is also mentioned by Grössel in his testimony. Conrad started his theological education in Cologne, where he was biblicus in 1431 and where he probably also read the Sentences. 16 He then transferred to Vienna, where he completed his theological studies with his last two exercises, his vesperies 17 on 4 June 1436 and, on his first day as master, his aula 18 disputation on 5 June 1436. 19 Thus Grössel refers to Conrad as a recent master when he died, with his final resting place in the Dominican convent of Vienna.

Another victim of the plague in 1436 was Johannes Stedler de Langhüta, who was a formed bachelor in Vienna at the time. He should not be confused with the Dominican Johannes Streler de Frankfurt, although they were contemporaries in the Faculty of Theology. 20 16 For a note on his life see I.W. Frank OP, Hausstudium und Universitätsstudium der Wiener Dominikaner bis 1500 (Archiv für Österreichische Geschichte 127), Graz 1968, pp. 217-218. 17 For the history of vesperies in medieval faculties of theology, see the forthcoming PhD dissertation of Luciana Cioca of Babes-Bolyai University: "The History of Vesperial Disputations in the 13 th and 14 th Centuries." 18 The history of the aula as part of the inception procedure in medieval universities is the subject of Nadège Corbiere's on-going PhD at EPHE: "Devenir maître en théologie au Moyen Âge : histoire de la cérémonie in aula." 19 The Acta Facultatis Theologie, p. 115, mentioned his aula under dean Iohannes Nider, for 5 June 1436: Item postea, feria 3 ante festum Corporis Christi, magister Cunradus Herbst habuit aulam suam et presentavit facultati II flor. ung. Frank, Hausstudium und Universitätsstudium der Wiener Dominikaner, p. 218, n. 16, also refers to these two academic exercises that are ending the process to obtain the title of doctor in theology in the former manuscript from the Dominikanerkloster from Basel, known today under the shelf mark: Basel, UB, UBH O III 45. This information is already vehiculed by G. Löhr, OP, Die Teutonia im 15. Jahrhundert. Studien existed. 22 Before starting his Sentences lectures he was already licensed in canon law. 23 From his efforts as cursor biblicus in the Faculty of Theology we have a commentary on Apocalypse from the New Testament and some Quaestiones in librum Iosuae from the Old Testament in manuscript Wien, Schottenstift, cod. 377 (Hübl 377), dated 1431, perhaps just after the completion of the lectures. [START_REF] Cf | Repertorium Biblicum Medii Aevi[END_REF] Stedler was admitted to read the Sentences in the fall of 1431 at the same time as Nicholaus de Grecz. 25 Testimony to Stedler's participation in the principia debates surfaces in manuscript Lilienfeld, Zisterzienserstift 85, described at length by Lackner [START_REF] Haidinger | The Handschriften des Stiftes Lilienfeld. Anmerkungen und Ergänzungen zu Schimeks Katalog[END_REF] and discussed recently by Ueli Zahnd. [START_REF] Zahnd | Plagiats individualisés et stratégies de singularisation. L'évolution du livre IV du commentaire commun des Sentences de Vienne[END_REF] I can now add to their analyses by identifying the author of the anonymous set of four principia on ff. 450r-481v of the manuscript with Nicholas de Grecz, who read the Sentences in Vienna in 1431-1433. This attribution is based on the references that the author makes to his socius, Johannes Stedler. For example, at the beginning of the second principium on the second book of the Sentences, Nicholas introduces his socius as follows: Tertio conferam cum magistro meo reverendo magistro Johanne Stredler in theologia baccalario cum quo licet immeritus in lecturam Sententiarum concurro. [START_REF] Ms | [END_REF] In addition, the biblical thema of the four principia sermons is Luke 21, 15: Ego enim dabo vobis os et sapientiam. At the time, the thema of the principia sermons was usually inspired by the first biblical book that the author was assigned to read as cursor, and in the case of Nicholas we know that this was the ewangelium beati Luce. 29 According to Grössel, in 1436 Stedler was a formed bachelor at the Collegium Ducale, meaning that he had read at least two books of the Sentences and performed his third principium -although we know from Nicholas that Stedler completed all four books and principia -but that he had not yet been licensed in theology, let alone promoted to master. 30 Finally, Grössel praises the memory of two other members of the Collegium Ducale who died of the plague. No indication of their identity is given and it is difficult to make any conjecture about them. One possible candidate is Master Jodocus Singer, who also died in 1436, perhaps by plague, as Aschbach suggests, 31 but about whom little else is known.

Grössel's insertion of this note at the end of his questions on Book III of the Sentences in an autograph manuscript illustrates the violence of this visit of the plague and the depth of its impact on the dynamics of the university: classes were suspended, scholars sent home, and colleagues, professors and friends lost for eternity. Testimonies on the plague overlap with the documentary history of the University of Vienna. Scholars, whether bachelors or doctors, beginners (indocti) or fully established in their profession, were equal before the medical emergency. Faced with the tremendous loss of as many 70 scholar in one day, the university decided to cancel all activities. Once a sort of normality returned, all disciplines resumed lecturing on the same day.

In the first half of the fifteen century the university was often confronted with this situation. Given our present circumstances and our recent experience with Covid-19, we are in now a better position to be sympathetic with Grössel's emotions. Seventeen years after the event recorded by Grössel, one of his fellows, the famous professor Thomas Ebendorfer, added a marginal note on 8 October 1453 signed on the last folio of manuscript Wien, ÖNB, 3369 informing the reader that he was checking texts of Petrarch and Boccaccio contained in the codex, since classes were suspended because of the plague: Revisus per Thomam de Haselpach tempore suspensionis lectionum ob sementem pestem. Anno etc. 1453, 8 Octobris.

Appendix

This appendix contains a transcription of the colophon in Wien, ÖNB, 4497, f. 266r. 32 The material condition of the codex is fragile, some words are difficult to be deciphered, and at the bottom of the folio a few words are illegible, so I indicate these words with xxx. The transcription follows the orthography of the codex and reproduces the names as they are written.

A

  STUDENT'S NOTES ON THE PLAGUE IN CODEX WIEN, ÖSTERREICHISCHE NATIONALBIBLIOTHEK, 4497 * Monica Brînzei (CNRS/UBB, Aubervilliers/Cluj-Napoca)

  Not much is known about Stedler, apparently from Landshut, northeast of Munich, but textual traces survive from his activities in the Faculty of Arts, where he was dean in 1433-1434, for example some disputed questions on De generatione et corruptione, 21 and his questions on Physics once in Wagendorfer, "Ein Bücherstestat an das Wiener Schottenstift," p. 274. For Gottlieb, Mittelalterliche Bibliothekskataloge Österreichs, p. 436, l. 26-29: Item quidam egregius Sacre Theologie doctor et magister, Urbanus de Melico nomine, legavit nobis ex singulari devocione septem bona volumina, que sibi erant cariora. Obiit autem prenominatus doctor quinta Ydus Augusti. Obiit autem prenominatus doctor quinto Ydus Augusti sepultusque est in ecclesia sancti Stephani, cuius canonicus erat et sacerdos. Wagendorfer, "Ein Bücherstestat an das Wiener Schottenstift," pp. 264-265.

15

See Wagendorfer, "Ein Bücherstestat an das Wiener Schottenstift," p. 274. This date is also confirmed by the Liber Oblationym et anniversariorum (1442-ca. 1480):

  und Texte vornehmlich zur Geschichte ihrer Reform (Quellen und Forschungen zur Geschichte des Dominikanerordens in Deutschland 19), Leipzig 1924, pp. 168-171. A detailed unpublished description of this manuscript by Gustav Binz, Basel 15. April 1937. See: https://swisscollections.ch/Record/991170499601505501.

20

See recently as a ERC Thesis project output P. Blazek, "Due commenti di Johannes Streler, O.P. (1459) alle Sentenze di Pietro Lombardo. Edizione dei prologhi e tabula quaestionum," Angelicum 91 (2014), pp. 669-726. 21 This texts are transmitted in ms. Melk, StB 802 as it is indicated on the entry of C.H. Lohr, "Medieval Latin Aristotle commentaries. Authors: Johannes de Kanthi-Myngodus," Traditio 27 (1971), pp. 251-351, here p. 283.

  Explicit scribendo pariter et legendo in profesto sancti Nicolai scriptum huiusmodi et lectum in 3° libro Sententiarum anno Domini M°CCCC36, cuius libri prefati inchoavi leccionem feria secunda immediate post octavas Pasce Anno 36. Attamen invalescente pestilentia anno predicto, videlicet 36, adeo quod de certo die una sepeliebantur 70 funera ante Assumptionis 29 Nicholaus de Grecz was admitted as a biblical cursor in October 1428, as we read in the AFTh, p. 63: Item petiverunt magistri Stephanus de Esslinga et Nicolaus de Grecz admitti ad legendum cursus, et fuerunt exauditi, et primo, scilicet magistro Stephano, fuit assignatus liber Sapientie, secundo vero ewangelium beati Luce. Geschichte der kaiserlichen Universität zu Wien, vol. 2, Wien 1854, p. 110, discussed by W. J. Courtenay, "Theological Disputations at Vienna in the Early Fifteenth Century. Harvard, ms. lat. 162," Bulletin de philosophie médiévale 53 (2011), pp. 385-401, here 386. Beate Virginis, me tunc Wyenne constituo, pluribusque notabilibus suppositis universitatis pro tunc et paulo ante defunctis, videlicet professore Sacre pagine Magistro Petro Pirchenbart in collegio Ducis seniore regente ac lectore in theologia; preterea Magistro Chünrado Herinbst similiter doctore, licet novo et non unius anni, in theologia, ordinis Predicatorum, sepulto in domo Predicatorum Wyenne. Ceterum Magister Urbanus de Mellico ecclesie sancti Stephani canonicus et in theologia doctore, peste violentante vita fungi desiit. Aliis omissis tam magistris, baccalariis quam scolaribus, viris moralissimis ac virtuosissimis, quorum felices cineres requiescant in summo. Postremo preceptis nature obtemperans debitumque eius persolvens Magister Johannes Strädlare Langhüta baccalareus formatus in theologia ac collegiatus in Collegio Ducis epidimia tactus expiravit in loco prescripto, puta collegio. Duobus etiam scolaribus magistrorum ibidem paucis intervenientibus diebus xxx ex post turbulentum mare presentis seculi exierunt. Quibus attentis universitas solemniter congregata indoctis et magistris conclusit ut decetero, videlicet a tempore paulo ante Assumptionis Marie, omnes actusscolastici, orationes pariter et consistoria universitatis penitus non fierent usque ad festum Cholomanni, aut si expediret et videretur per amplius huiusmodi actus suspenderentur. Quo concluso et habita plena universitatis vacatione, recessi post responsionem meam in aula statim ad partes nativas, die videlicet dominica immediate ante Assumptionis absentando me tredecem integris septimanis et die una. Postremo Wyennam redii feria tertia post Elizabeth inveniens tantum quatuor collegiatos presentes adhuc, cum quibus ego quintus. Toto enim tempore a suspensione lectionum et post recessum meum dumtaxat unus magistrorum collegio preerat, omnibus aliis absentibus, usque ad Cholomanni. In festo autem Katherine congregabatur universitas in suppositis ac doctoribus ac magistris, et pro tunc eligebantur primo omnes universitatis officiales et eadem die xxx xxx assignabantur lectiones in omnibus facultatibus.

30

On the significance of baccalareus formatus see R. Kink, 31 Aschbach, Geschichte der Wiener Universität, p. 476: "Der Magister Jodocus Singer dagegen stirbt schon 1436, wahrscheinlich und der Pest."

32

The manuscript is described in various catalogues; see supra n. 00.

H. Göhler, Das Wiener Kollegiat-, Nachmals Domkapitel zu Sankt Stephan in Wien 1365-1554, Wien-Köln-Weimar 2015, pp. 368-369.

 5 See Aschbach, Geschichte der Wiener Universität, pp. 588-592.

Gottlieb, Mittelalterliche Bibliothekskataloge Österreichs, p. 481, reproduces an interesting note on this treatise still existing in 1467: Item magister Iodocus de Ravenspurg et dominus Georgius sacerdos, testamentarii olim magistri Iohannis de Kupferberg, presentaverunt decano facultatis pro tempore quosdam libros facultati arcium legatos pro debito duorum talentorum ex parte census, quem obligabatur predictus magister facultati. Tituli vero librorum sunt isti: Questiones super libris metaphisice. Disputata Stedler in libris phisicorum. Questiones magistri Petri de Walse super libris Ethicorum. Item quosdam sermons inconcilio pronunciatos et primi thema 'Erat signa in sole et luna'. Item liber continens tractatum per modum dyalogi


Arnaud Sternchüss
email: arnaud.sternchuss@ecp.fr

Etienne Balmès

Pierrick Jean

Jean-Pierre Lombard

Reduction of multi-stage disk models: Application to an industrial rotor

The present study deals with the reduction of models of multi-stage bladed disk assemblies.

The proposed method relies on the substructuring of the rotor into sectors. The bladed disks are coupled by intermediate rings which remove the problem of incompatible meshes. The sectors are represented by super-elements whose kinematic subspaces are spanned by a set of cyclic modeshapes and a set of normal modes when their left and right interfaces are fixed. The first step is to compute the cyclic modeshapes that are defined on the full rotor by enforcing the uncoupling of the spatial Fourier harmonics. This leads to a family of subproblems parameterized by the harmonic coefficient, similarly to the classical approach used to deal with tuned bladed disks. The subsequent reduction process leads to compact reduced models whose accuracy has been extensively tested on simple but realistic academic models. The proposed methodology was then applied to an industrial rotor to conduct an analysis at a wider scale. This case was also the occasion to point out the fact that the assembly of individual disk models into a rotor model is really straightforward and provides an efficient tool to observe and predict coupled phenomena.

INTRODUCTION

This study presents an industrial application of the authors' reduction technique to determine the modal characteristics of assemblies of bladed disks.

The current industrial practice is to model separately each bladed disk of the rotor. When such disks are cyclically symmetric, their dynamics are fully determined from that of a small portion, typically a bladed sector. This configuration no longer holds in real rotors due to mistuning which significantly alters this behaviour since it can strongly localize the strain energy 1 . Therefore, full-scale analyses require huge models, all the more since many nearly identical computations are needed to cover the entire space of parameters (mistuning coefficients, rotation speed, temperature,...). To address this issue, many reduction techniques have been developed such as those of Lim et al. 2 , Feiner and Griffin 3,4 and Sternchüss et al. 5 , among others, that rely on synthesis techniques using either component or system modes.

Such reduced representations would be a good starting point for dealing with multi-stage rotors. However, as underlined by Bladh et al. [START_REF] Bladh | Effects of Multistage Coupling and Disk Flexibility on Mistuned Bladed Disk Dynamics[END_REF] , the critical point is the choice of the boundary conditions that would properly represent inter-stage coupling, which is the main cause of discrepancies between single stage and multi-stage dynamics, since this coupling ratio controls the spread of the strain energy over many disks. Song et al. 7 , Laxalde et al. 8 , and Sternchüss and Balmès [START_REF] Sternchüss | Reduction of Multistage Rotor Models Through Cyclic Modeshapes[END_REF] proposed recently efficient CMS techniques that include such coupling directly into the formulation of the reduced problem.

The first part of this paper updates the methodology developed in [START_REF] Sternchüss | Reduction of Multistage Rotor Models Through Cyclic Modeshapes[END_REF] . This method consists in a decomposition of the disks into sector super-elements connected with each other through inter-sector elements and inter-disk ring elements. The updates mostly concern the formulation of the problems that lead to the cyclic modeshapes as defined by Laxalde et al. 8 , and the process used to generate the subspaces in which the kinematics of the sector super-elements are sought. The assembly of the problem remains really straightforward, leading to a very compact model whose accuracy is driven by the choice of the reduction bases. The main results obtained for a simple academic model that evaluates the accuracy of the proposed technique are described throughout this section.

The second part of this paper then presents the application of the methodology to a subset of a realistic industrial rotor provided by Snecma. From these results, the concluding section provides some guidelines for the practical use of such a reduction method in a global

Coupled mechanical problem

The meshes of the aft rim I 1 a of D 1 and the front rim I R f of R are identical to avoid problems of incompatibility. For the same reasons, the meshes of the aft rim I R a of R and the front rim I 2 f of D 2 are identical. Due to the way it is meshed, the inter-disk ring R has a single element along the z direction, thus it has no interior DOF when it is made of first order elements.

{q 1 a } = {q R f } and {q 2 f } = {q R a } (1)
The particular geometry of R implies that its DOF are fully determined by {q R } =

{ q R f q R a } ⊤ .
In the frequency domain, if an external load {f e (ω)} is given, the discrete displacement {q(ω)} satisfies:

[Z(ω)] N ×N {q(ω)} N ×1 = {f e (ω)} N ×1 (2)
The dynamic stiffness matrix [Z] of the rotor is

[Z(ω)] =   Z 1 (ω) 0 0 Z 2 (ω)   + Z R (ω) (3)
where [Z 1 ], [Z R ] and [Z 2 ] stand for the dynamic stiffness matrices of D 1 , R and D 2 respectively.

Spatial harmonic content

Let us consider a single disk D d in its nominal configuration. Under the assumption of cyclic symmetry, it is made of N d s identical angular sectors, numbered from 0 to N d s -1, generated by successive rotations of a reference sector. Let θ d be the rotation of angle α d = 2π/N d s around the z axis. Since the discrete geometry is invariant whatever the rotation (θ d ) s , the discrete displacement q d,s is paired with its Fourier harmonic coefficients q d δ such that, in the local basis of sector S d,s , it is written 5,10 :

∀s ∈ N N d s -1 , q d,s = ℜ   N d s /2 δ=0 q d δ e i s δ α d   (4)
The spatial harmonic DOF vectors q d δ are defined on the reference sector S d,0 and are complex quantities, except for δ = 0 and δ = N d s /2, if N d s is even, multiplied by a dephasing factor that depends on the harmonic coefficient δ and the sector number s. Throughout the rest of the paper, the terms associated with harmonic coefficient N d s /2 are systematically kept, even if N d s is odd, in order to avoid dealing with different cases. As stated by Eqn. (4), there exists a duality between the physical DOF vector q d and the harmonic DOF vector q d :

q d = q d,0 q d,1 q d,2 • • • q d,Ns-1 ⊤ q d = q d 0 ℜ q d 1 ℑ q d 1 • • • ℜ q d N d s /2-1 ℑ q d N d s /2-1 q d N d s /2 ⊤ (5)
ℜ and ℑ stand for the real and imaginary part of a complex quantity. The relation between these DOF vectors is

q d = E d ⊗ I N d,0 q d (6)
where [I N d,0 ] is the identity matrix whose size is the number of physical DOF of sector S d,0

and ⊗ is the Kronecker product.The block-column [E d δ ] that corresponds to coefficient δ is

[E d 0 ] = [ 1 • • • 1 • • • 1 ] ⊤ [E d δ ] =   2 2 cos(δ α d ) • • • 2 cos(s δ α d ) • • • 2 cos((N d s -1) δ α d ) 0 -2 sin(δ α d ) • • • -2 sin(s δ α d ) • • • -2 sin((N d s -1) δ α d )   ⊤ [E d N d s /2 ] = [ 1 • • • (-1) s • • • (-1) N d s -1 ] ⊤ (7)
When dealing with an assembly of two disks, the assumption about the ring states that any physical DOF belongs either to disk D 1 or to disk D 2 . Therefore, the global DOF vector

{q} is {q} =   E 1 ⊗ I N 1,0 0 0 E 2 ⊗ I N 2,0   { q} (8)
where {q} = { q 1 q 2 } ⊤ and { q} = { q 1 q 2 } ⊤ .

Thanks to this, the spectral content of the modes of the whole rotor can be determined relatively to each disk. With the previous diagram, some particular modes can be identified, as described in Fig. 3. The harmonic DOF vectors are here normalized such that max

d∈{1,2} max δ∈{0,N d s /2} q d δ 1 = 1 (10)
One then finds mono-harmonic coupled modes, like the near-pair of modes at f = 4.25 and f = 4.26 that seems to correspond to δ = 2. Thus, the ring participates in theses modes, but its lack of symmetry does not affect the motion very much. On the contrary, the mode at f = 2.99 which is localized to disk D 1 is multi-harmonic and the perturbation lies in the asymmetry of the ring, even though its participation is small.

Mono-harmonic eigensolutions

From the previous study of the complete rotor, the main conclusion to be drawn is that modes are not always mono-harmonic. According to this, mono-harmonic solutions such

Paper number GTP-08-1085 8,[START_REF] Laxalde | Analyse Dynamique des Structures Cycliques Multi-Étages[END_REF] are not necessarily modes of the full rotor. In the following section, the theory associated with mono-harmonic solutions is recalled and the error between them and the eigenmodes is qualified.

Let us consider two disks D 1 and D 2 connected with a volumic interface R without interior DOF as presented in Fig. 1. The physical displacement is substituted by its Fourier decomposition (Eqn. ( 8)) in the coupled problem in Eqn. (2). If one considers that any disk

D d , d ∈ {1, 2}, is tuned, its dynamic stiffness [Z d ] is obtained from that of its reference sector [Z d,0 ] through [Z d ] = [I N d s ⊗ Z d,0 ].
E d ⊤ ⊗ I N d,0 ] is [ Z d ] = [E d ⊤ ⊗ I N d,0 ] [I N d s ⊗ Z d,0 ] [E d ⊗ I N d,0 ] = [E d ⊤ E d ] ⊗ [Z d,0 ] =           N d s Z d,0 0 • • • 0 0 0 2 N d s Z d,0 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • 2 N d s Z d,0 0 0 0 • • • 0 N d s Z d,0           (11)
Thanks to Eqn (11), the projection of the block-diagonal matrix containing the dynamic stiffnesses of both disks in Eqn.

(3) is really straightforward:

  E 1 ⊗ I N 1,0 0 0 E 2 ⊗ I N 2,0   ⊤   Z 1 0 0 Z 2     E 1 ⊗ I N 1,0 0 0 E 2 ⊗ I N 2,0   =   Z 1 0 0 Z 2   (12) [ Z R ] is defined such that [ Z R ] =   E 1 ⊗ I N 1,0 0 0 E 2 ⊗ I N 2,0   ⊤ Z R   E 1 ⊗ I N 1,0 0 0 E 2 ⊗ I N 2,0   (13)
If [Z R ] is decomposed into blocks indexed by dd ′ that couple disks D d and D d ′ , the blocks

indexed by dd ′ in [ Z R ] are then [ Z R dd ′ ] = [E d ⊗ I N d,0 ] ⊤ [Z R dd ′ ] [E d ′ ⊗ I N d ′ ,0 ] (14)
The different blocks indexed by dd ′ can themselves be decomposed into blocks indexed by δδ ′ that couple a shape of harmonic δ on disk D d to a a shape of harmonic

δ ′ on D d ′ : [ Z R dd ′ ,δδ ′ ] = [E d δ ⊗ I N d,0 ] ⊤ [Z R dd ′ ] [E d ′ δ ′ ⊗ I N d ′ ,0 ] (15)
This inter-harmonic coupling prevents the mechanical problem from being decomposed into individual subproblems for every harmonic coefficient as it could be done in the mono-disk analysis. This also implies that the free vibration modes of the rotor can only be monoharmonic if the inter-harmonic coupling coefficients in (15) have a negligible influence. In the present formulation, the submatrices indexed by δδ ′ , δ = δ ′ , in the blocks [ Z R dd ′ ] are neglected, so that harmonics are decoupled.

[ Z R dd ′ ] = [ Z R dd ′ ,δδ ] + X X X X X [ Z R dd ′ ,δ =δ ′ ] (16)
Paper number GTP-08-1085

9 A. Sternchüss With this approximation, [ Z R δ ] is defined retroactively such that [ Z R δ ] =   E 1 δ ⊗ I N 1,0 0 0 E 2 δ ⊗ I N 2,0   ⊤ [Z R ]   E 1 δ ⊗ I N 1,0 0 0 E 2 δ ⊗ I N 2,0   (17)
The approximation error can be quantified from [ Z R dd ′ ,δ =δ ′ ]. In future work, this error estimation will be integrated into a residual iteration technique in order to enrich the subspaces of approximation in the subsequent reduction process.

From the structures of the harmonic matrices in Eqns. ( 12) and ( 17), one obtains a class of parametrized problems with respect to δ:

• if δ = 0     N 1 s Z 1,0 0 0 N 2 s Z 2,0   + [ Z R 0 ]      q 1 0 q 2 0    = {0}   c 1 l -c 1 r 0 0 c 2 l -c 2 r      q 1 0 q 2 0    = {0} (18) • else        2        N 1 s Z 1,0 0 0 0 0 N 1 s Z 1,0 0 0 0 0 N 2 s Z 2,0 0 0 0 0 N 2 s Z 2,0        + [ Z R δ ]                     ℜ( q 1 δ ) ℑ( q 1 δ ) ℜ( q 2 δ ) ℑ( q 2 δ )              = {0}   c 1 l -c 1 r e 2 i π δ α 1 0 0 c 2 l -c 2 r e 2 i π δ α 2      q 1 δ q 2 δ    = {0} (19)
The equations that come first in the problems above are not sufficient to define a cyclic modeshape. An observation equation that states how the shapes are dephased between two adjacent sectors is added to each problem. The observation matrices [c d r ] and [c d l ] are used to isolate the right and left interface DOF.

In the second problem, the projection of the dynamic stiffness matrix of any of the disks is equivalent to duplicate this matrix. This is not the case for the ring, which leads to nearpair eigensolutions. For thinner meshes of the ring, the near-pair solutions would converge to exact pairs. Throughout the rest of this paper, only the solutions without aliasing are considered: for the eigensolutions of the rotor or the modes of the individual disks when considered free-free. This graph shows significant discrepancies which can be explained by the fact that the inter-stage interfaces are not exactly fixed when the disks are assembled together, but they are not free either. Accurate boundary conditions to apply to individual disks would be somewhere between free and fixed, depending on the modeshape.

δ ≤ min

Model Reduction

According to the previous section, mono-harmonic solutions are not modes, but they will be used to approximate the kinematics of the bladed sectors, in order to build a reduced model of the rotor. To do so, each bladed sector is represented as a super-element that allows:

(i) to target a set of eigenmodes from a set of mono-harmonic solutions. Since it is assumed that each eigenmode is the linear combination of mono-harmonic solutions whose frequencies lie in the vicinity of the corresponding natural frequency, the targeting of an eigenmode is in fact the targeting of the narrow frequency range where its harmonic components are found. It is also possible to select a subset of the harmonics a modeshape is made of. In practice, only the mono-harmonic solutions with low δ are selected, since only these modeshapes account for disk-dominated motion at low frequencies 1 .

(ii) to include blade-dominated motion into the reduced model by adding a set of modes of the individual sectors with their left and right interfaces clamped. Such modeshapes are used since fewer of them are required to account for blade-dominated motion than cyclic modeshapes with high δ at low frequencies.

Degrees of freedom of the super-elements

Three types of generalized DOF are assigned to each sector super-element, they are defined in correspondance to a subset of physical DOF:

(i) DOF that belong to the inter-sector elements (green area in Fig. 6),

(ii) DOF that belong to the inter-disk ring (red area in Fig. 6),

Paper number GTP-08-1085 (iii) purely interior DOF (the other DOF).

Generalized DOF sets (i) and (ii) can be joint. In the previous approach 9 , since the gen-

Subspaces of motion

Let us consider a sector S d,s with:

(i) a set of eigenmodes [Φ d,s f ix ] of this sector with its left and right interfaces both clamped. These modes can differ from a sector to another.

(ii) a set of mono-harmonic solutions [ ℜ([Φ d,s rcyc ]) ℑ([Φ d,s rcyc ]) ] whose harmonics δ are selected for a particular reason (design, ...). These modeshapes are identical for all the sectors of a given disk.

The initial set of vectors is:

[T d,s init ] = [ [Φ d,s f ix ] ℜ([Φ d,s rcyc ]) ℑ([Φ d,s rcyc ]) ] (20)
The sorting order of these vectors is critical to ensure a good condition number of the

{q d,s } = [T d,s M ] {q d,s M } (21)
Even though it is never explicitely built, the subspace in which the approximate solutions

Paper number GTP-08-1085 14 A. Sternchüss are sought is given by Eqn. ( 22).

[

T M ] =              T 1,0 M • • • 0 0 • • • 0 . . . . . . . . . . . . . . . 0 • • • T 1,N 1 s -1 M 0 • • • 0 0 • • • 0 T 2,0 M • • • 0 . . . . . . . . . . . . . . . 0 • • • 0 0 • • • T 2,N 1 s -1 M              (22)
and the generalized DOF vector {q M } is related to the physical DOF vector {q} thanks to Eqn. (23).

{q} = [T M ] {q M } ( 23
)
Assembly of the super-elementary matrices

Instead of computing the reduced matrices by the direct projection

[Z M ] = [T M ] ⊤ [Z] [T M ]
where [Z] is defined in Eqn.

(3), a two-step procedure is followed:

Step 1: the physical finite element matrices of the sectors are projected onto their own kinematic subspace:

[Z d,s M ] = [T d,s M ] ⊤ [Z d,s ] [T d,s M ] (24)
[Z d,s ] is the dynamic stiffness of sector S d,s . These reduced matrices are then assembled with respect to the generalized DOF associated with the super-elements. This operation leads to the reduced matrices of both sector and inter-sector super-elements.

Step 2: the matrices of the ring are projected onto the kinematic subspaces of the neighbouring disks with respect to the inter-disk generalized DOF.

[Z R M ] = [T M ] ⊤ [Z R ] [T M ] (25)
In order to avoid using the full matrix of modeshapes [T M ], [Z R M ] is built iteratively sector by sector. All the reduced matrices are relatively sparse. Besides, they have the remarkable form displayed in Fig. 8 for the assembly of two tuned disks with 12 and 15 blades. The diagonal blocks and four off-diagonal blocks derive from the projection of the sector matrices. The latter blocks can be divided into two groups, the first with 12 blocks and the second with 15 blocks. The inter-disk coupling is contained both in the diagonal blocks that are derived from the matrices of the inter-sector joints and in the off-diagonal blocks that are derived from the matrices of the ring. Apart from these remarks, the reduced matrices have no other properties which could impact their eigenvalues or eigenvectors. Notice that the generalized load can be obtained from the physical load with the same procedures.

Paper number GTP-08-1085 16 A. Sternchüss

Numerical Results

Once the reduced matrices and load vectors have been assembled, it is possible to compute the free or forced response of the rotor.

The generalized eigenmodes are solutions of the homogeneous problem:

[Z M (ω)]{q M (ω)} = {0} (26)
The physical modeshapes are then restored to the full structure thanks to Eqn. (23). These modeshapes can be compared to those computed with the full model.

In the case of the sample rotor with 12 and 15 blades, different sets of mono-harmonic solutions with δ ∈ {0, 5} were computed for various targeted frequency ranges and were used to reduce the rotor problem, as displayed in Fig. 9a. 250 generalized modes, comprised between f = 0 and f = 15, were compared to the assumed corresponding modes of the rotor in the same frequency range, referred to as reference modes. The resulting relative error in frequency is plotted in Fig. 9b. A modeshape correlation index, based on a Modal Assurance Criterion (MAC) is reported in Fig. 9c. As can been seen in these figures, the maximum of the relative error in frequency is below 0.1% and the modeshape correlation is excellent as soon as subspace [T M ] includes enough vectors. A future development of this work is to define a priori error estimators, probably based on the identified source of error underlined in the mono-harmonic computation section.

APPLICATION TO AN INDUSTRIAL MODEL

The industrial model to which the methodology described in the previous section is applied is a part of the numerical model of a compressor in development, it is depicted in geometries of the sectors of two connected stages are such that they are bonded together, a pre-processing of the geometries is required to slightly translate the nodes of the flanges of disks D 1 and D 3 so that the inter-disk rings can fill the empty spaces created between the flange of disk D 1 (resp. disk D 3 ) and the rim of disk D 2 , such as depicted in Fig. 11. Two sets of nodes have to be moved for each flange:

• the first set is translated longitudinally,

• the second set is translated radially.

This operation does not affect the integrity of the finite element mesh of such parts. Notice

Paper number GTP-08-1085 18 A. Sternchüss that the bolting system is not included in the model. The minimal set of elements to From Fig. 13, it is noticed that coupled solutions are found inside a triangle that has δ = 0, f = 0 as a summit, and δ = 0 and f = δ as edges (when f is normalized, as stated before). In this region of high to moderate disk coupling as δ increases, there are

Paper number GTP-08-1085 19 A. Sternchüss noticeable differences between the frequencies of the rotor solutions and the disk natural frequencies. This comes from the fact that a fix condition at inter-stage interfaces is too stiff and disk modes that are similar to solutions of the rotor are higher in frequency. On the contrary, modes of the disks with free rims would be below rotor solutions since a free condition at the inter-stage interfaces is far too soft. Ideal boundary conditions to apply to a mono-disk analysis so that multi-stage effects are properly described would be between free and fix condition, and they would vary according to the considered diameter solution. (i) a small disk-blade coupling, which tends to confine the strain energy on the blades 1 . In this case, the global motion is very similar to that of the individual disks whatever their boundary conditions are, since the latter do not affect the blade-dominated motion due to the low coupling ratio between the disk and the blades.

(ii) a small disk-disk coupling, which tends to confine the strain energy on a single disk.

Besides, a phenomenon of aliasing occurs for mono-harmonic solutions that have a strong participation of disk D 1 (resp. D 2 ). Typical modeshapes are plotted in Figs. 14 to 17.

Notice that the computation of the strain energy in Fig. 14b provides useful information since the maximum of strain energy is located at the rim between disks D 1 and D 2 and not at the tip of the blades as expected because of the 2S1 shape.

Paper number GTP-08-1085 The participation factor on each disk, in the sense of Eqn. ( 27), is plotted in parallel to these spectra.

∀d ∈ {1, 2, 3}, R d = {q d M } ⊤ [K d M ] {q d M } {q M } ⊤ [K M ] {q M } (27)
It is mandatory to examine both graphs together to avoid hasty conclusions in terms of multi-harmonicity of modes. Indeed, a mode that looks multi-harmonic on a given disk can be so if its participation relative to that disk is very small: the analysis of an almost zero signal leads to a spectrum that shows multiple harmonics, but it does not indicate that the generalized displacement is almost zero on this disk! Special attention must be paid to the fact that a physical indicator, such as the kinetic energy, should be used to avoid normalization problems between the kinematic subspaces defined on each disk. For instance, modes 118 -180 are clearly localized to disk D 3 according to Fig. 18 but seem to be multi-harmonic on disks D 1 and D 2 . The only multi-harmonic modes of the reduced model are observed on disk D 3 for δ > 22 (if δ is considered as the dominating harmonic in these particular cases), as described in Fig. 21 where the shape of mode number 428 at f = 10.23 is restored to the whole assembly, and its weighted spectrum in the sense of Eqn. ( 10) is plotted. From the latter figures, it can be concluded that almost all modes are Paper number GTP-08-1085 22 A. Sternchüss mono-harmonic, like that of Fig. 22.

Recovery of Modeshapes and Post-processing

Post-processing the restored modeshapes on the full rotor would be a very expensive numerical effort, since storing the 900 generalized modes represents about 305Mb, whereas storing the 900 modes once restored represents 53Gb. A procedure of partial recovery of the modeshapes has been developed. Given a selection of elements where the quantities of interest (energies, stresses) are assumed to be located, the generalized response is restored to the physical DOFs of these elements only. These locations can be given:

• either by a spatial Fourier analysis of the generalized displacement or the strain energy associated with each super-element, which leads to recoveries to full sector models,

• or if the generalized displacement appears to be mono-harmonic, a procedure of pairing between mono-harmonic solutions and generalized modes (not described here) returns useful information where to look for stresses or energies.

As an illustration, the generalized mode at f = 15.89 can be manually paired with the mono-harmonic solution with δ = 1 at f = 15.53 depicted in Fig. 14. Figure 23 displays the strain energy and the Von Mises stress distributions on a subset of physical finite elements that belong to disk D 1 . This figure shows that, for this generalized mode and this disk, the Von Mises stress is maximum on the disk and, as expected, has a high value in the middle of the blade tip due to the 2S1 shape. Nevertheless, the mono-harmonic solution has a strong participation of disk D 2 and a recovery extended to that disk should be computed.

Paper number GTP-08-1085

N

  Number of degrees of freedom N s Number of sectors R Participation factor in strain energy The considered structure is presented in Fig. 1. It is composed of two bladed disks nominally tuned D 1 and D 2 with 12 and 15 blades respectively connected with a volumic interface R. The disks are meshed with brick elements and the ring is meshed with tetrahedrons obtained by a Delaunay tessellation (MATLAB function delaunay3) between the nodes of the aft rim of disk D 1 , denoted I 1 a , and the front rim of disk D 2 , denoted I 2 f . The material is titanium whose properties are E = 120 GPa, ν = 0.33 and ρ = 4700 kg.m -3 .The radii at the roots of the blades are 200 mm and 205 mm respectively, the radii at their tips are 280 mm and 285 mm respectively. Their thickness varies from 6 mm at the root to 2 mm at the top. Their widths are 43 mm and 46 mm at the top. The frequency bandwidth of interest for this study is [0, 14], if all frequencies are normalized with respect to the lower flexible natural frequency of the rotor.

  FIG. 1: Sample assembly

  FIG. 2: Spatial spectrum of the 74 first flexible modes of the sample rotor

  FIG. 3: (a) Mono-harmonic coupled mode at f = 4.25 and (b) multi-harmonic localized mode at f = 2.99 and their harmonic content

  The projection of the latter matrix onto the subspace Paper number GTP-08-1085 8 A. Sternchüss generated by the Fourier matrix [

2

  FIG. 4: Near-pair of mono-harmonic solutions with δ = 2 at f = 4.25

FIG. 6 :

 6 FIG. 6: Sector model: ( ) sector super-element, ( ) inter-sector super-element and ( ) inter-disk ring surface elements

FIG. 7 :

 7 FIG. 7: Reduced rotor model: ( ) sector super-elements, ( ) inter-sector super-elements and ( ) inter-disk ring elements

FIG. 8 :

 8 FIG. 8: Reduced stiffness matrix [K M ] (a) global shape: ( ) blocks of disk D 1 , ( ) blocks of disk D 2 and ( ) blocks of ring R, and (b) actual matrix The reduced dynamic stiffness [Z M ] is finally assembled with respect to the full set of generalized DOF.

FIG. 9 :

 9 FIG. 9: (a) Targeted frequency ranges, (b) frequency relative error and (c) modeshape correlation between reference and generalized modes.

Fig. 10 .

 10 Fig. 10. It is composed of three integrally bladed disks (blisks), whose characteristics are reported in Tab. I. The full finite element model of the assembly is made of 11, 289, 230 tetrahedrons connecting 2, 631, 094 nodes, which leads to 7, 893, 282 DOF. The material of the stages are special alloys of titanium. Since the finite element model is quite huge as is, only linear elements were considered at the time of the present study, but computations with quadratic elements are planned.As described in the beginning of this paper, one uses inter-disk rings built from a Delaunay tessellation between the nodes of the regarding rims of two adjacent disks. Since the nominal

FIG. 11 :

 11 FIG. 11: Details of the rings (a) between D 1 and D 2 and (b) between D 2 and D 3

FIG. 12 :Fig. 13 .

 1213 FIG. 12: Sector models and rings used in the computation of cyclic modeshapes (a) front and (b) rear

FIG. 13 :

 13 FIG. 13: Normalized frequencies of the cyclic solutions for the rotor ( ) and the modes of disk D 1 (+), disk D 2 (×) and disk D 3 (+ ×) in free-fix, fix-fix and fix-free condition. (-), (-) and (-) are frequencies of single sectors with left and right interfaces fixed

FIG. 14 :FIG. 16 :

 1416 FIG. 14: Coupled cyclic solution with δ = 1 at f = 15.53: (a) modeshape and (b) density of strain energy. Blade shapes are 2S1, 2F and 1T.

FIG. 19 :

 19 FIG. 18: Generalized modes 7 to 300 (f ∈ [1, 7.52]): (a) Spatial harmonic content (b) Participation factor in strain energy for disks ( ) D 1 , ( ) D 2 and ( ) D 3

  FIG. 20: Generalized modes 601 to 900 (f ∈ [13.76, 19.57]): (a) Spatial harmonic content (b) Participation factor in strain energy for disks ( ) D 1 , ( ) D 2 and ( ) D 3

FIG. 21 :FIG. 22 :

 2122 FIG. 21: Restored mode at f = 10.23 localized to disk D 2 : (a) modeshape and (b) spatial harmonic content for disks ( ) D 1 , ( ) D 2 and ( ) D 3

FIG. 23 :

 23 FIG. 23: Partial post-processing of the generalized mode at f = 15.89 on disk D 1 : (a) normalized density of strain energy and (b) normalized distribution of Von Mises stress

TABLE I :

 I Description of the finite element model

	Stage Nb. of Nb. of	Nb. of	Nb. of Type of
		sectors Nodes/sec. Elements/sec. DOF/sec. Elements
	1	36	16, 434	65, 170	49, 302	Tetra4
	2	56	9, 566	36, 815	28, 698	Tetra4
	3	70	23, 104	97, 860	69, 312	Tetra4

Paper number GTP-08-1085

A. Sternchüss

Acknowledgments

Thanks go to Snecma for its technical and financial support. This work takes place in the framework of the MAIA mechanical research and technology program sponsored by CNRS, ONERA and SAFRAN Group.

CONCLUSION

In the present paper, the authors updated then applied the reduction technique they developed to deal with multi-stage assemblies of bladed disks [START_REF] Sternchüss | Reduction of Multistage Rotor Models Through Cyclic Modeshapes[END_REF] to a realistic rotor model.

The results obtained for this industrial rotor allow to give some guidelines to use efficiently the previous method in a real industrial environment:

(i) In the proposed methodology, a set of mono-harmonic solutions are computed. Such solutions already contain much valuable information in terms of coupling or regions of high density of strain energy or high stress levels.

(ii) These solutions are used to build a compact reduced model that represents the multistage model with a controlled accuracy. In future work, the error estimator derived from the mono-harmonic computations will be integrated into an iterative procedure of enrichment of the subspaces built to approximate the kinematics of the sectors.

(iii) On the one hand, if a spectral analysis shows that the generalized modes of interest are spatially mono-harmonic, it means that they potentially correspond to monoharmonic solutions, given by a manual or automatic pairing process. This property will be exploited in future work, especially in forced response computations.

(iv) On the other hand, if some of the modes of interest are multi-harmonic, the response is computed with the reduced model.

(v) In every case, the response can be restored to a subset of elements chosen either from an a priori analysis of the mono-harmonic solutions or from an a posteriori analysis of the spatial spectrum of the generalized modes.

The proposed methodology is already implemented within the Structural Dynamics Toolbox [START_REF] Balmès | Structural Dynamics Toolbox 6.0 (for use with MATLAB)[END_REF] with critical use of the super-element capabilities of this library. Internal element formulations are used although imports from SAMCEF and NASTRAN have also been done for verification. The SDT switch to use NASTRAN as the eigenvalue solver for SDT based models was sometimes used for very large models. The SDT/OpenFEM export to VTK format was used to generate some of the figures included in this paper with ParaView.

Developments specific to this study, grouped as the embryo of an SDT/Rotor package, are Paper number GTP-08-1085 29 A. Sternchüss (i) pre-processing tools to assemble sector models and mesh rings using an automated volume mesher, (ii) mono-harmonic rotor computations, (iii) sector super-element building from a set of mono-harmonic multi-stage solutions and fixed sector modes, (iv) post-processing tools for harmonic and generalized coordinate viewing.