domain: sdv shs

text:
Ugo Arbieu
email: ugo.arbieu@senckenberg.de

Kathrin Helsper

Maral Dadvar

Thomas Mueller

Aidin Niamir

Corresponding Author

Natural Language Processing as a tool to evaluate emotions in conservation conflicts

Keywords: Artificial Intelligence, Conservation conflicts, Emotion Detection, Framing effects, Humancarnivore coexistence, Media Content Analysis, Sentiment analysis HIGHLIGHTS

Conservation conflicts involve an important emotional component that has been little investigated so far. In particular, the study of emotions involved in conservation conflicts have been limited in their scope (e.g. negative vs. positive sentiments) by the language studied (mostly English). Natural Language Processing (NLP) approaches combined with informed expert knowledge constitute a promising tool to analyze the occurrence of discrete emotions in textual contents. We applied NLP on the textual content of German print and online news publications featuring the return of wolves (Canis lupus) in the Saxony region of Germany (n=3,096), to investigate the occurrence of 7 basic emotions and their associations with 9 stakeholder groups. All emotions could be detected with different accuracy levels, and except for interest that was detected in all articles, negative emotions tended to dominate the articles' emotional content (anger in 74% and fear in 36% of the articles). Anger was most frequently detected in articles featuring farmers (20%) and hunters (15%), as was fear (17% with farmers, 14% with hunters). Our study of a diverse set of emotions, in local language, demonstrates the usefulness of NLP approaches to broaden the understanding of local conservation conflicts, in which the news organizations have a pivotal, yet often neglected role.

INTRODUCTION

Conservation conflicts and the underlying tensions between stakeholders over wildlife management can generate heated debates [START_REF] Chapron | The place of nature in conservation conflicts[END_REF][START_REF] Nyhus | Human-wildlife conflict and coexistence[END_REF], involving strong emotions. The role of emotions in conservation conflicts has been recognized in recent years [START_REF] Chapron | Recovery of large carnivores in Europe's modern human-dominated landscapes[END_REF][START_REF] Jacobs | Understanding emotions as opportunities for and barriers to coexistence with wildlife[END_REF][START_REF] Jacobs | Human emotions toward wildlife[END_REF][START_REF] Nelson | Emotions and the ethics of consequence in conservation decisions: Lessons from Cecil the lion[END_REF], as they influence our dispositions towards wildlife [START_REF] Jacobs | Understanding emotions as opportunities for and barriers to coexistence with wildlife[END_REF] and seem tied to attitudes [START_REF] Landon | Cognitive and affective predictors of Illinois residents' perceived risks from gray wolves[END_REF][START_REF] Sponarski | The role of cognitions and emotions in human-coyote interactions[END_REF][START_REF] Straka | Understanding the acceptability of wolf management actions: roles of cognition and emotion[END_REF]. However, they remain seldom studied. To date, studies on emotions towards wildlife have mostly focused on fear [START_REF] Flykt | Fear of wolves and bears: Physiological responses and negative associations in a Swedish sample[END_REF][START_REF] Johansson | Is human fear affecting public Willingness to Pay for the management and conservation of large carnivores?[END_REF]. The current lack of knowledge on emotions in the context of conservation probably owes to the lack of consensus around the definition of emotions [START_REF] Izard | Basic emotions, natural kinds, emotion schemas, and a new paradigm[END_REF][START_REF] Jacobs | Human emotions toward wildlife[END_REF], their diversity [START_REF] Jackson | Emotion semantics show both cultural variation and universal structure[END_REF] and their cultural and language specificities [START_REF] Aman | Identifying expressions of emotion in text[END_REF]. The study of emotions in the context of conservation conflicts thus requires strong interdisciplinary integrations and appropriate tools to address these challenges.

In recent years, Natural Language Processing (NLP) approaches, i.e. Artificial Intelligence techniques that allow computers to read and interpret information from natural language texts [START_REF] Thessen | Applications of Natural Language Processing in biodiversity science[END_REF], have been broadly used in multidisciplinary biodiversity projects [START_REF] Thessen | Applications of Natural Language Processing in biodiversity science[END_REF]. Recent studies have highlighted the usefulness of Artificial Intelligence methods in conservation, as they can be used to identify conservation icons [START_REF] Ladle | Conservation culturomics[END_REF], track illegal wildlife trade on social media [START_REF] Di Minin | Machine learning for tracking illegal wildlife trade on social media[END_REF][START_REF] Xu | Use of machine learning to detect wildlife product promotion and sales on Twitter[END_REF], investigate human-nature interactions [START_REF] Toivonen | Social media data for conservation science: A methodological overview[END_REF] or sentiments towards iconic species like white rhinoceros [START_REF] Fink | Online sentiment towards iconic species[END_REF]. We argue that NLP approaches, , combined with informed expert knowledge [START_REF] Dadvar | Experts and machines against bullies: A hybrid approach to detect cyberbullies[END_REF][START_REF] Hirschberg | Advances in natural language processing[END_REF][START_REF] Hovy | What are sentiment, affect, and emotion? Applying the methodology of Michael Zock to sentiment analysis[END_REF][START_REF] Mohammad | Sentiment analysis: Detecting valence, emotions, and other affectual states from text[END_REF] constitute a promising avenue to investigate emotions in conservation conflicts, as they can be applied to a broad set of textual resources, to a wide range of contexts, and most importantly, to datasets containing an amount of data that cannot be processed manually. However, to date, automatic detection of emotions in textual content (i.e. sentiment analysis) applied in conservation have been limited by (i) the dichotomy between positive and negative emotions, although emotions are more diverse than simple polarizations (see Hausmann et al. 2020 for a recent illustration in South African National Parks), and (ii) the focus on English language resources, which is, in many cases, of limited local conservation relevance.

In this study, we applied NLP in automatic analysis of emotions on the textual content of print and online news publications featuring the return of wolves (Canis lupus) in the Saxony region of Germany, where they have been present since their recolonization of Germany in 2000 [START_REF] Reinhardt | Military training areas facilitate the recolonization of wolves in Germany[END_REF]. Wolf conservation is one of the most contentious conservation topics in human-dominated landscapes [START_REF] Darimont | Political populations of large carnivores[END_REF], and its return to Germany has sparked important media coverage. Stories involving returning carnivores are good candidates for being selected by news organizations, as they are perceived to be relevant to a broad readership (although few people interact with carnivores), involve a charismatic animal and often feature negative news (e.g. depredation events, concerns for human health) [START_REF] Harcup | What is news? Galtung and Ruge revisited[END_REF]. Hence, human-carnivore coexistence is frequently featured in the press [START_REF] Bombieri | Content analysis of media reports on predator attacks on humans: Toward an understanding of human risk perception and predator acceptance[END_REF][START_REF] Fernández-Gil | Conflict misleads large carnivore management and conservation: Brown bears and wolves in Spain[END_REF][START_REF] Killion | Tracking a half-century of media reporting on gray wolves[END_REF] and contains an important emotional component that play an important role in the mechanisms behind the effect of news frames on opinions, or framing effects [START_REF] Lecheler | The mediating role of emotions[END_REF]. Here, we investigated a set of seven basic emotions, namely interest, surprise, joy, sadness, fear, anger and disgust [START_REF] Jacobs | More than fear: role of emotions in acceptability of lethal control of wolves[END_REF]. In addition, we investigated the co-occurrence of the seven emotions and nine specific wolf conservation stakeholders, namely politics, research, Non-Governmental, Organizations, farmers, hunters, military, news agencies, media, Saxony's Wolf Information Office.

MATERIALS and METHODS

Corpus of news

Since wolf return to Saxony in 2000, the Wolf Information Office "Kontaktbüro Wölfe in Sachsen" has been continuously compiling press clippings featuring wolves, in print and digitized formats. The sampling protocol has evolved through time: at first, articles were collected manually by surveying local (i.e., from the region of Saxony) and national press, and opportunistically collecting news on wolves from other federal states; then, the advent of online newspapers allowed automatized data collection based on keyword entries. It is therefore possible that some articles were missed, but we assume that this would have mostly affected articles from sources outside Saxony, and we do not anticipate specific biases in terms of topics or emotional content. We obtained 4,000 newspaper articles, all in German, that covered the entire period from 2000 to 2018, to investigate their content. As most articles were indeed published in Saxony (n=2,556) or were from national sources (n=764), we excluded articles from German federal States other than Saxony and focused on a set of 3,320 newspaper articles. Each article was scanned and processed with an Optical Character Recognition program. We only extracted the textual information and excluded articles containing unrecognized text (n=224), resulting in a total of 3,096 articles for our analysis.

Natural Language Processing

We pre-processed the texts using the Natural Language Toolkit in Python: we removed stop words and non-informative symbols and punctuations, and used stemming to reduce them to their word stem. The identification of emotions followed several steps (see Supporting Information Fig. S1). First, we developed our own lexicons to identify the seven emotions in the articles' texts, by compiling words referring to each of the seven emotions. We used the SentimentWortschatz (V1.8) dictionary [START_REF] Remus | SentiWS -a publicly available German-language resource for sentiment analysis[END_REF] to further enhance our list and find synonyms to the words associated with each of the seven emotions (see Supporting Information Table S1). Second, we used the Term Frequency-Inverse Document Frequency (TF-IDF) numerical approach [START_REF] Aizawa | An information-theoretic perspective of tf-idf measures[END_REF] to identify the most informative and distinct words in each text and to investigate whether they included the words related to the emotions. Third, we identified the presence of emotions in each text based on the frequency of occurrence of words related to a specific emotion (Whitelaw et al., 2005).

Fourth, we used a word-level n-gram model (here bigram, n=2) that takes into account the co-occurring words within a given window around the words associated with the specific emotions. This provided the possibility to further look into the context that a word has been used for, rather than only the frequency of its occurrences.

Stakeholder categorization and extraction

We created a lexicon for each of the stakeholder groups by populating a list of the keywords based on our own expertise. We further improved the lexicon using an inductive approach and adding terms that we found in the newspapers during the labelling process (see Evaluation). We included politics, research, Non-Governmental Organizations, farmers, hunters as these groups are most active in wolf conservation debates. We further included military (military training areas have been critical for wolf recolonization in Germany) [START_REF] Reinhardt | Military training areas facilitate the recolonization of wolves in Germany[END_REF], news agencies, media (i.e. people involved in cultural and artistic activities associated with wolves e.g. books, films, exhibits, etc.) and the Wolf Information Office as these were frequently mentioned in the articles. We then used the lexicons as a proxy to detect the occurrence of the stakeholders in each article (see Supporting Information Table S2, Fig. S1).

Evaluation

The evaluation dataset consisted of 498 articles manually and independently labelled for presence/absence of the seven emotions by two annotators with a social-ecological background. The inter-annotator agreement was 0.80 (average agreement over the 7 emotions) meaning that in 80% of the labelled cases, the reviewers labelled the newspaper articles to the same emotion categories (see Table 1). We used the manually labeled dataset, where both annotators were in agreement, to evaluate the accuracy of the NLP approach and to report the precision and the recall of the identified emotions. For a specific emotion, precision is the fraction of accurately detected instances (true positives) among the detected instances by the NLP (true positives + false positives), whereas recall is the fraction of accurately detected instances (true positives) among the total amount of relevant instances, i.e. texts that genuinely contained that specific emotion (true positives + false negatives). Accuracy also reported the F1-Score which is the number of correctly identified emotions divided by the number of all correctly identified emotions returned. Finally, to investigate the co-occurrence of emotions and stakeholder groups in the newspaper articles, we used pairwise Pearson correlation tests between the seven emotions and the nine stakeholder groups, and implemented a bonferroni correction on p-values to account for multiple tests.

RESULTS

The classification accuracy was different among emotions (Table 1). Disgust (1.00) and interest (0.84) had the highest precision whereas joy (0.43) and surprise (0.62) had the lowest. Interest (1.00) and anger (0.61) had the highest rates of recall whereas disgust (0.08) and joy (0.09) had the lowest rates. Hence, interest (0.91) and anger (0.63) had the highest F1-scores, whereas disgust (0.14) and joy (0.15) had the lowest (Table 1). Across 3,096 newspaper articles on wolves, interest (1.00), anger (0.74) and fear (0.36) were the most frequently detected emotions (Fig. 1).

We identified politics (0.56) and farmers (0.54) as the most frequently found stakeholders in the articles (Fig. 1). Comparatively, the Wolf Information Office (0.04) and the military (0.04) were the least featured stakeholders in the articles (Fig. 1). The most frequently detected co-occurrences between emotions and stakeholders in the newspaper articles were between farmers and anger (0.20, t=11.33, p < 0.001), farmers and fear (0.17, t=9.68, p < 0.001), hunters and anger (0.15, t=9.68, p < 0.001) and hunters and fear (0.14, t=7.82, p < 0.001) (Fig. 2). The highest correlation between two emotions was found between anger and fear (0.36, t=21.57, p < 0.001).

DISCUSSION

Main findings

Overall, our approach using NLP to detect emotions in German articles featuring wolves demonstrated that all emotions could be detected in the corpus of 3,096 articles, and that the approach could be applied to another language than English, increasing its local relevance for conservation. Although emotion detection accuracy was sometimes low, we detected a domination of negative emotions like anger and fear. The detection of fear and anger in news articles featuring farmers and hunters, although reflecting the challenges of human-carnivore coexistence, may hint at an unbalanced representation of wolf conservation in the news.

Natural Language Processing' relative efficiency to detect emotions

Emotion detection is a challenging task because it is language-and topic-specific [START_REF] Aman | Identifying expressions of emotion in text[END_REF]. The inter-annotator agreement of ca. 80% illustrated the challenge of labelling texts containing specific emotions. However, this value was in the range of previous sentiment analysis in German [START_REF] Momtazi | Fine-grained German sentiment analysis on social media[END_REF] and other languages [START_REF] Mozetič | Multilingual Twitter sentiment classification: The role of human annotators[END_REF]. Furthermore, the F1-scores, reflecting quantity and quality of detected emotions, were also in line with previous findings [START_REF] Balahur | Detecting implicit expressions of emotion in text: A comparative analysis[END_REF]Kušen et al., 2017), reporting higher precision for negative emotions and lower recall values for positive emotions (Table 1), meaning that negative emotions are more accurately detected, and positive emotions occur less frequently in the news. Such pattern probably echoes the news tendencies to select and frame negative news more frequently [START_REF] Harcup | What is news? Galtung and Ruge revisited[END_REF]. Interest may not be a relevant emotion as it was detected in all newspaper articles, which publication probably already reflects an expression of interest, thereby artificially inflating recall values (1.00).

Except for interest, the prevalence of negative emotions like anger, fear and sadness reflected the overall negative framing of human-carnivore coexistence in the news [START_REF] Bombieri | Content analysis of media reports on predator attacks on humans: Toward an understanding of human risk perception and predator acceptance[END_REF][START_REF] Nanni | Social media and large carnivores: Sharing biased news on attacks on humans[END_REF]. The domination of anger and fear is problematic for carnivore conservation, as emotions play an important role in mediating news framing effects on people's attitudes [START_REF] Lecheler | The mediating role of emotions[END_REF]. In fact, it is widely accepted that negative information has a stronger impact on people's attitudes than positive information [START_REF] Soroka | Good news and bad news: Asymmetric responses to economic information[END_REF]. As only few people directly interact with wolves [START_REF] Arbieu | The positive experience of encountering wolves in the wild[END_REF] or other conflict-prone wildlife, human-wildlife relations are mainly shaped by information and not personal experience. Therefore, at a time where human activities increasingly encroach on wildlife habitats and potentially generate more conservation conflicts, the conservation community should embrace the complexity of emotions in conservation conflicts, for longterm human-wildlife coexistence. We demonstrated that NLP approaches are promising tools in this regard.

Emotions and stakeholders' representation

News organizations play an important role in giving a voice to conservation actors, and stakeholder recognition is a critical aspect of conservation conflict management [START_REF] Redpath | Don't forget to look down -collaborative approaches to predator conservation[END_REF]. We are aware of only few studies investigating stakeholder occurrence in the media (see e.g. [START_REF] Rust | Media framing of financial mechanisms for resolving human-predator conflict in Namibia[END_REF][START_REF] Webb | Conversations in conservation: revealing and dealing with language differences in environmental conflicts[END_REF]. Our approach using NLP to investigate the co-occurrence between stakeholder and emotions revealed that the highest associations between hunters, farmers, and anger and fear depicted a bleak image of human-wolf coexistence in Saxony. This finding is at odds with a previous study that found rather neutral attitudes expressed by the broad public towards wolves in this region [START_REF] Arbieu | Attitudes towards returning wolves (Canis lupus) in Germany: Exposure, information sources and trust matter[END_REF], reflecting an attenuation of the conflict over time. This discrepancy between the negative and emotional content found in the newspapers that focus on the short-term, often negative news and people's rather neutral attitudes that reflect longer timeframes of learning to coexist with carnivores, illustrates the news agencies' narrow focus and perception of human-carnivore coexistence. Understanding stakeholder's as well as news agencies' perceptions [START_REF] Bennett | Using perceptions as evidence to improve conservation and environmental management[END_REF]) may be critical to restore a more balanced and fair representation of human-carnivore coexistence in the media. In addition, the prevalence of politics and the under-representation of research demonstrates that the information conveyed by news organizations may not always be evidence-based, and calls for improved communication from the scientific community. Therefore, NLP approaches can be a useful tool to investigate local conservation conflicts and potential polarizations [START_REF] Webb | Conversations in conservation: revealing and dealing with language differences in environmental conflicts[END_REF]. Besides, they are useful to analyse big textual datasets and resources, such as online social media platform contents and news archives. However, most existing tools and technical resources, such as training datasets and gold standards for automatic detections have mainly been developed for the English language. Therefore, having these tools and classification models developed for other languages, such as our study in German, could be beneficial to local conservation contexts. In particular, this will help identifying imbalances in stakeholder representation, investigating emotional contents, and developing strategies to produce evidence-based information that do not mainly focus on short-term negative news.

Outlook

This work calls for further investigations of the role of emotions in news featuring conservation topics. First, to improve the accuracy of emotion detection, we would recommend manually labeling an evaluation dataset that covers sufficient numbers of each emotion to train a machine learning model, prepared by at least 3 annotators. Second, further investigations could go beyond emotion occurrence and address emotional intensity, and investigate changes at different spatial and temporal scales. Such investigations would be very important in contexts of rapid social-ecological changes. Third, comparisons across charismatic species could inform on discrepancies in species framings [START_REF] Bombieri | Content analysis of media reports on predator attacks on humans: Toward an understanding of human risk perception and predator acceptance[END_REF]. Most importantly, we encourage conservation scientists to embrace interdisciplinarity and develop collaborations with scholars across journalism studies, natural and computer sciences.

Figure 1 .

 1 Figure 1. Distribution of emotions (A) and stakeholders (B) across 3,096 press articles reporting on the wolf topic in Saxony, Germany.

Figure 2 .

 2 Figure 2. Correlations between emotions (pink) and stakeholders (green) across 3,096 pressarticles reporting on the wolf topic in Saxony, Germany. Non-significant correlations (5% threshold, p-values adjusted with bonferroni correction) are blanked. We excluded interest from the plot, as it was detected in all newspaper articles (recall = 1.00, see Fig.1). The size of the circles represents the correlation strength.

Table 1 .

 1 Annotators' categorization (among Ntotal = 498), inter-annotator agreement and classification accuracy (precision, recall and F1-scores) of the automatically identified emotions across 498 newspaper articles from Saxony, Germany

		Annotator 1	Annotator 2
		categorization	categorization	Inter-annotator
		(N articles)	(N articles)	agreement	Precision Recall	F1-score
	Interest	402	269	0.72	0.84	1.00	0.91
	Anger	275	235	0.86	0.65	0.61	0.63
	Fear	220	162	0.83	0.76	0.41	0.53
	Sadness	189	143	0.80	0.68	0.34	0.46
	Surprise	209	100	0.72	0.62	0.13	0.21
	Joy	185	150	0.85	0.43	0.09	0.15
	Disgust	147	67	0.81	1.00	0.08	0.14

Acknowledgements

The authors thank the members of the Kontaktbüro "Wölfe in Sachsen" for the collection of newspaper articles and S. Liehn for coordinating the data handover; we thank the members of the Lupus Institute for Wolf Monitoring and Research including I. Reinhardt and G. Kluth for their support; we thank A. Renthrop, E. Bauer and A. Höfer for their help in designing the emotion lexicons and annotating the articles to create the evaluation dataset. This study was supported by the Robert Bosch Foundation

AUTHOR CONTRIBUTION

UA, AN and TM conceptualized the manuscript. UA, AN and MD developed the methodology. MD developed the Natural Language Processing approach. KH collected the data from the press articles. UA, AN and KH performed the analyses. UA prepared the results visualization. UA prepared the initial draft of the manuscript and all authors contributed substantially to the redaction of the final version.

Supporting Information Natural Language Processing as a tool to evaluate emotions in conservation conflicts

Ugo Arbieu 1,2* , Kathrin Helsper 1,3 , Maral Dadvar 4 , Thomas Mueller 1,5 , Aidin Niamir 1 1 Senckenberg Biodiversity and Climate Research Centre, Senckenberganlage 25, 60325 Frankfurt am Main, Germany.

2 Smithsonian Conservation Biology Institute, National Zoological Park, 1500 Remount Road, Front Royal, VA 22630, USA.

3 Zoological Institute, Christian-Albrechts-Univerität zu Kiel, Am Botanischen Garten 1-9, 24118 Kiel, Germany. * corresponding author: ugo.arbieu@senckenberg.de


Bendali Salhi
email: bendali.salhi@univ-fcomte.fr

Phd Student

Professor Marc Berthillier
email: marc.berthillier@univ-fcomte.fr

Professor Joseph Lardies
email: joseph.lardies@univ-fcomte.fr

Philippe Voinis
email: philippe.voinis@edf.fr

Charles Bodel
email: charles.bodel@edf.fr

A SUBSPACE METHOD FOR MODAL IDENTIFICATION OF BLADED ASSEMBLIES USING BLADE TIP-TIMING DATA

Keywords: blade vibration analysis, blade tip-timing method, subspace method

Unknown excitation forces are applied to bladed disk assemblies, such as turbines blades, leading to forced vibration responses. Non contact measurement of such vibrations using blade tip-timing data has become an industrial standard procedure and current research focuses on analysis methods for interpretation of measured vibrations. Our purpose is to develop a method for identification of the blade's natural frequencies and damping ratios using blade tip-timing data. The method is based on a subspace analysis. Its performance is compared to the traditional Fast Fourier Transform analysis. A detailed description of these methods and results are presented.

INTRODUCTION

Bladed assembly vibration measurements are crucial to the prediction of blade life. Generally, blade vibrations in compressor or turbine stages during operation are difficult to measure and require many sensors in order to be fully detected [START_REF] Dimitriadis | Blade tip timing measurement of synchronous vibrations of rotating bladed assemblies[END_REF]. Two standard measuring methods are used to measure blade vibrations: strain gages and frequency-modulated Grid Systems. The first method consists of installing strain gages on some of the blades. This is a very expensive process since it requires slip rings or high-quality telemetry and long installation times. For the second standard method, the frequency modulated Grid Systems (FM grid), permanent magnets are fitted on the tips of some blades and a specially formed wire is installed in the compressor casing above the magnets. As a blade passes a wire, the magnet produces an alternating current (AC) in the wire. Blade vibrations are interpreted from the voltage modulation of this AC signal. No signal transmitter between the rotor and the stator is needed, but the wire requires a special casing and the number of blades monitored at a time is limited to three per rotor stage. An alternative method to the two standard measurement methods is the blade tip-timing data (BTT) or non intrusive stress measurement which is based on optical, capacitive or magnetic probe technology.

Fig.1. A SIMPLE BLADE TIP-TIMING SYSTEM

This paper describes the use of subspace modeling for identification of blade vibration frequencies and their damping ratios using blade tip-timing data. This method is applied to both: a lumped parameter model which produces asynchronous responses and to real measurements of a turbine in operating conditions.

BLADE TIP-TIMING METHOD

The basic concepts of this technique have already been known for many years [START_REF] Carrington | A comparison of blade tip timing data analysis methods[END_REF][START_REF] Craig | Turbo-machinery blade vibration amplitude measurement through tip timing with capacitance tip clearance probes[END_REF][START_REF] Dimitriadis | Blade tip timing measurement of synchronous vibrations of rotating bladed assemblies[END_REF]. Several probes are installed in the engine casing above the rotor, and the blade transit times between the probes are measured. Figure 1 summarizes the basic principle of this technique. Two sensors are placed around the blades. They measure the passage time of each blade in front of the sensors. The angle between these two sensors is of 180°. A third sensor is placed in the proximity of the engine shaft. It measures the once-per-revolution signal.

When there are no blade vibrations, the blade transit times are a function of rotor speed, rotor radius, and circumferential probe position. In the case of blade vibrations, the blade transit times deviate from those in the undisturbed condition, with the blades passing the probes earlier or later than normal, depending on their instantaneous deflection. Compared with the two standard methods cited previously, this technique provides notable advantages: 1) it is non-contacting; 2) it senses all blades; and 3) it reduces costs because it eliminates the need for rotor instrumentation and telemetry for signal transmission. This technique however, has some drawbacks. One such drawback is that the measurement sampling frequency is completely dependent of the rotor speed and the number of measurement probes installed.

The signals presented in figure 2 are the once-perrevolution signal and the passage time of seventy-seven blades in front of the two sensors. The difference between the actual time of arrival TOA (the blade arrival time at the front of the sensor) of the vibrating blade and its computed TOA (when it has not been in vibration) provides the raw data from which the instantaneous blade displacement is found. These displacements are found in the following way: in each revolution the values of the once-per-revolution data and the average value of the rotation speed are removed from raw data to find blade vibration times. To obtain blade displacements, the vibration time of each blade is multiplied by the value of rotation speed.

Methods for analysing blade tip-timing data have been developed since 1970. However, given a limited number of probes, there are still no standard approaches that can identify synchronous response resonance frequencies with adequate precisions. Let us recall that synchronous response occurs when a blade's vibration frequency is an integer multiple of the engine rotation speed and asynchronous response occurs when the blades frequency response is a non-integer multiple of the assembly rotation speed. Synchronous vibration can be caused by mechanical effects such as residual unbalance of the rotors and non-concentric casing, as well as aerodynamic effects such as irregular pressure distributions within the airflow due to the engine intake geometry and wakes produced by upstream stators. Asynchronous vibrations are typically caused by flutter, rotating stall, or by acoustic resonance.

The analysis of blade tip-timing data can be performed using on-line data analysis which provides near real-time information updated at least once per second or by off-line data analysis. In this article we consider off-line analysis to obtain more detailed information on individual blades.

THE SUBSPACE METHOD

The subspace method is based on the use of a state space model which is obtained by transforming equation of motion for the bladed assemblies: state equation and an observation equation. The discrete time state space model is given by [START_REF] Lardies | Estimation of parameters and model order in state space innovation forms[END_REF][START_REF] Lardies | A stochastic realisation algorithm with application to modal parameter estimation[END_REF][START_REF] Zhang | Modal parameter identification using response data only[END_REF]:

) (t f Kq q C q M = + + into a
+ = + k 1 k k z Az Be state equation (1) k k k e Cz y + = observation equation (2)
where k y is the (m×1) vector of observation, k e the vector of stochastic innovations, k z the (n×1) vector of states, A the (n×n) transition matrix, B the (n×m) Kalman gain matrix, and C the (m×n) observation matrix. The matrices A, B, C and the order n of the multivariate stochastic process are parameters to be estimated, from output data only.

In order to estimate these parameters, the shifted observability matrix method presented here is used. Let + k y be the (mf×1) future data vectors and k y the (mp×1) past data vectors such that:

+ + + - = ' ' ' ' k k k 1 k f 1 y ( y , y ,..., y ) (3)
- - -+ = ' ' ' k k k 1 k p 1 ' y ( y , y ,..., y ) (4)
The covariance matrix between the future and the past observations is the (mf×mp) Block Hankel matrix given by:

+ - + - + + - = =             1 2 p ' 2 3 p 1 k k 1 f f 1 f p 1 R R . R R R . R H E[ y y ] . . . . R R . R (5)
where E denotes the expectation operator, H is the Block Hankel matrix (a block band counter diagonal matrix) formed with the (m×m) individual covariance matrices:

- + = = ' i 1 i k i k R E[ y y ] CA G (6) with + = ' k 1 k G E[ z y ] . The sample covariance matrices are computed from the data k y , k=1, 2…T ∑ - = + - = 1 T 1 k ' k i k 1 i y y T R ˆ; i=0, 1…p+f (7)
and are used to estimate a sample Block Hankel matrix Ĥ . In order to identify the model coefficients, two factorizations of the Block Hankel matrix are considered. The first factorization uses the orthogonal-triangular (QR) decomposition of Ĥ as: = Ĥ QR with R an upper triangular matrix of the same dimension as Ĥ and Q an identity matrix. The second factorization of the Block Hankel matrix considers its (mf×n) observability and (n×mp) controllability matrices, O and K as [START_REF] Lardies | Estimation of parameters and model order in state space innovation forms[END_REF][START_REF] Lardies | A stochastic realisation algorithm with application to modal parameter estimation[END_REF]:

- - = =                 p 1 f 1 C CA H G AG A G OK . CA (8)
The two factorizations of the Block Hankel matrix are for a given order n:

= = n n n n n ˆˆĤ Q R O K (9)
To estimate A, it is necessary to introduce the following shifting matrices

↓ - =             f 2 C CA O . CA and ↑ - =             2 f 1 CA CA O . CA (10)
where,

↓

O is the (m(f-1)×n) matrix obtained by deleting the last block line matrix of the observability matrix O and ↑ O is the (m(f-1)×n) matrix obtained by deleting the first block line matrix of the observability matrix O. It is easy to show that the transition matrix A is related to these two matrices by the following expression:

↑ ↓ = O O A (11)
Using generalized inverses we deduce the transition matrix as:

↓ ↑ =     # A O O (12)
It is easy to show that the observation matrix C is obtained directly from the observability matrix O by considering the first block line of O:

C = O 1L = {first block line of O} (13)
Once the transition matrix A has been estimated its eigenvalues λ i and eigenvectors φ λ are used to identify the modal characteristics of the system: eigenfrequencies i F , damping ratios i ξ and mode shapes Φ :

λ λ λ λ π∆ λ λ + - = +                     2 2 * * ln( ) i i 1 1 i i F cos i * 2 t 4 2 i i (14) λ λ ξ λ λ λ λ λ λ = + - +                         2 * ln( ) i i i 2 * 2 * 1 i i ln( ) 4 cos i i * 2 i i (15) φ Φ C = (16)

VALIDATION METHODOLOGY

This section describes the methodology used to develop and verify efficiency of the subspace technique. Figure 3 shows the two degrees of freedom per sector model. This model simulates the dynamic behavior of specific bladed assemblies which has seventy-seven blades, and provides a temporal forced response of each blade. The mechanical and aerodynamic parameters used in this model were obtained by a theoretical mode shape analysis. Afterwards, the subspace method [START_REF] Lardies | Estimation of parameters and model order in state space innovation forms[END_REF][START_REF] Lardies | A stochastic realisation algorithm with application to modal parameter estimation[END_REF][START_REF] Zhang | Modal parameter identification using response data only[END_REF] was employed to identify the modal parameters of the structure: eigenfrequencies and damping ratios by using the temporal forced response of each blade. To verify the efficiency of the identification technique, the results obtained will be compared with the model's exact data found by a simple modal computation. Figure 4 presents the flowchart of the proposed methodology.

MECHANICAL MODEL

The mechanical model used in this study and shown in figure 1, has blade (d o f) and disk (d o f) for each sector. Each disk (d o f) is coupled with neighbouring sectors by springs of stiffness K c . This model simulates the dynamic behavior of a bladed assembly which has seventy-seven blades. The mechanical and aerodynamic parameters used in this model were obtained from a theoretical mode shape analysis. The external efforts are simulated by a random excitation and structural damping is assumed to be present.

Mistuning is modeled by adding stiffness variations to nominal blade stiffness. The i th mistuned stiffness is expressed as:

) 1 ( , i b i b k k δ + = (17)
where, δ i is the i th mistuning value. In this study δ i is given using experimental data.

1 f f 2 mean i i -         = δ (18)
where f i is the frequency of the i th blade and f mean is the mean frequency of all blades.

Aeroelastic coupling has been introduced, in the way of circular matrix coefficients, considering only five different coefficients. The equations of motion for the model are: where, q is the vector of displacements, M is the mass matrix, C is the damping matrix and K is the stiffness matrix. Matrix details are provided in the appendix A.

) (t f Kq q C q M = + + (19)
The eigenvalues of the tuned and mistuned model are presented in figure 5. These computed eigensolutions will be called exact solutions in the rest of the paper. The tuned system is unstable, due to the presence of negative damping coefficients, while the mistuned system is stable. Mistuning couples all the tuned modes and stabilizes the modes unstable.

MATHEMATICAL MODEL

The system described by the equation ( 19) was presented in the state space of 2N dimension (N is the number of (d o f)).

 

    +            - =             0 ) ( 0 0 0 t f q q M K q q M M C (20)
The temporal forced response is then given by:

∫ - - + = t t T t S T t t S d g Y Xe Ux Y Xe t x 0 0 ) ( ) ( ) ( 0 ) ( τ τ τ (21)
where Y, X and S are respectively the left eigenvector matrix, the right eigenvector matrix, and the diagonal matrix of eigenvalues (using the usual mathematical notations).

This method of computation gives the vibration signal of each blade in a negligible computing time compared to methods of temporal integration (Newmark and Runge Kuta tested on the same system). The second advantage is that this method gives exact results because the differential equation ( 20) is solved without approximation. Figure 6 presents a simulated temporal forced response of one blade. The simulation was made over 24 minutes with a sampling step 0.007 seconds.

Our purpose is to identify the eigenfrequencies and damping ratios of the blades using only such temporal responses.

SIGNAL RECONSTRUCTION

The subspace method requires signals without aliasing to identify modal parameters, whereas tip timing data gives signals with aliasing. To solve this problem and find a link between the subspace method and tip-timing data, we propose in this paragraph an interpolation technique based on the Shannon sampling theorem [START_REF] Ries | Digital time-delay beam forming with interpolated signals[END_REF]. This theorem is given by the following expression:

) k t ( c sin ) k ( s ) t ( s s k s complex _ analytic - = ∑ +∞ -∞ = ν ν (22)
where k is the number of samples, sinc is the sinus cardinal function, and s analytic_complex is the complex signal of the real signal s aliasing (signal with aliasing) given by tip-timing data. The Hilbert Transform is used to compute s analytic_complex from s aliasing . s analytic_real is the interpolated signal of bandwidth B and center frequency f 0 . Its expression is given by the following relation:

) ) ( ( 0 2 _ t f i real analytic e t s real s π × = (23)
An analytic band-pass signal with bandwidth B and center frequency f 0 can be perfectly reconstructed from samples of the corresponding complex envelope at a sub-sampling rate ν s > B.

Hence the sampling rate s ν can be much smaller than twice the highest frequency (f 0 +B/2). The following formula describes the reconstruction of this signal:

)] k t ( c sin e ) k ( s [ real s s ) k t )( / f ( 2 i k s complex _ analytic real _ analytic s s 0 - × × = - +∞ -∞ = ∑ ν ν ν ν π B s > ν (24)
Using this formula, the original high-frequency signals s analytic_real can be reconstructed at an arbitrary time samples, which is important for many applications, such as tip-timing data.

However, formula of this type are known to be impractical, since the sinc function decays very slowly, so that many terms of the series have to be evaluated for accurate reconstruction. A well known approach to avoid this problem in the complex envelope or low-pass case is to use reconstruction kernels K that are time-limited so that only a finite number of samples have to be considered. The general reconstruction formula, for s analytic_real sampled at rate s ν , is given by:

)] k t ( K e ) k ( s [ real : ) t ( s K s ) k t )( / f ( 2 i k s complex _ analytic real _ analytic s s 0 s - × × = - +∞ -∞ = ∑ ν ν ν ν π ν (25)
where, K is piecewise continuous function.

The optimum kernel for an arbitrary order m is given in the time domain by:

∑ - = - = ] 2 / ) [( 0 2 ) ( ) 1 ( ) )( ( l m l m l lm l t B D b t m Kopt (26)
where, B n is the B-spline function of order n. It is a piecewise polynomial of degree n-1 with finite support in [-n/2, n/2], defined recursively by:

( ) ] 2 / , 2 / [ 2 ) 1 ( )! 1 ( 1 ) ( 1 ] 2 / [ 0 1 n n t k t n k n n t B n t n k k - ∈ ×       - -         - - = - - = ∑ (27) and ) )( ( ) ( 1 1 t B B t B n n * = - (28)
is the n-fold convolution of the rect-function, which is equal to B 1 , with itself. The number b lm is defined by: ( )

∑ ∞ = =         0 2 2 / sin 2 / l l lm m t b t t (29)
The appendix B presents the optimum kernels used for the reconstruction of the sub-sampling signals.

Figure 7 presents the results obtained using the reconstruction formula (25).

In this figure the reconstructed signal (obtained from the signal with aliasing) is compared to the original signal (the same signal without aliasing). The reconstruction error is given by the 2-norm of difference value between reconstructed and original signal. The following relation describes how this error is computed:

2 original ted reconstruc S S Error - = (30)
for m=1, the error of reconstruction is equal to 6 10 -3 .

IDENTIFICATION OF MODAL PARAMETERS

Two approaches are proposed in the identification process using the subspace method described previously: global and local approach.

The global approach uses all temporal blade responses. This approach allows us to identify the mode shapes of the bladed assemblies. The local approach uses the temporal response of each blade alone. The results obtained, using the local approach, are presented here.

Figure 8 gives a comparison between the identified and the exact eigenfrequencies. The signal used in the identification process respects the Shannon theorem (no aliasing). The average quadratic error in the identified eigenfrequencies is 3.91. It is given by the following expression:

2 2 1 ∑ - = j j j J ω ω (31)
To check the efficiency of the subspace method, the results found are compared to those obtained though a standard discrete Fourier Transform. Figure 11 gives comparison between the identified eigenfrequencies, found by using FFT method, and the exact eigenfrequencies. Figure 12 presents the relative errors between the identified and the exact eigenfrequencies (the exact eigenfrequency for the i th blade is the eigenfrequency of the mode shape for which maximum displacement is obtained for this i th blade). The average quadratic error in the identified eigenfrequencies obtained by a standard Fourier transform is 44.78.

These results indicate that the subspace method is better suited for identifying closely spaced blade vibration frequencies than the FFT method.

The second advantage of the subspace technique is identification of the damping ratios. The random decrement technique [START_REF] Ibrahim | Random Decrement Technique for Modal Identification of Structures[END_REF] is used to extract the damped vibration from random responses of each blade. Afterwards the subspace method is used to identify the damping ratios of each blade. Figure 13 gives the damped vibration of blade n° 77. Figure 14 gives the relative errors between the identified and the exact damping ratios (the exact damping ratio for the i th blade is the damping ratio of the mode shape for which maximum displacement is obtained for this i th blade). We observe that the level of relative error is very high for 12 blades (more than fifty per cent). We are currently working to improve the identification of damping coefficients using a weighted subspace method.

CONCLUSION

A new analysis method of blade tip-timing data is proposed in this article. This technique is based on the subspace method using QR decomposition. The identified eigenfrequencies have been compared to those obtained by an FFT standard method. The comparison between the average quadratic errors of the two methods shows that the subspace method is more efficient than the standard FFT method: the average quadratic error in identified eigenfrequencies using the subspace method is 3.91 while it is of 44.78 using the standard FFT method.

It can be noted that the subspace method gives easily the damping ratios; these coefficients can not be identified by FFT method. However, the level of relative errors in damping ratios is remarkably high for certain blades. We are currently working to improve identification of the damping coefficients using a weighted subspace method.

Another advantage of the subspace method is the identification of the structure mode shapes. This identification method gives good results only if all temporal forced responses of the bladed assembly are simultaneously treated. However, identification using the subspace method requires long signals in the case of eigenfrequencies closely spaced. Consequently, computing time will be bulky and identification of mode shapes becomes impossible for bladed assemblies which have more than thirty blades. We are currently working to improve identification of mode shapes by a new technique based on a weighted subspace method. This technique allows identification of mode shapes in a reasonable computing time.

Despite the good results of the subspace method, in particular the identification of closely spaced eigenfrequencies, this method suffers from some disadvantages. One such disadvantage is that the subspace method, like others, is robust only if the signals are not aliased. To surpass this problem a

k k k k k k K       - = 3 0 0 0 " k K
reconstruction technique of the sub-sampled signal has been proposed. This technique provides good results and allows identification using the subspace method.

Fig. 2 .

 2 Fig.2. RAW BLADE TIP-TIMING DATA

Fig. 4 . 4 V

 44 Fig.3. TWO DEGREES OF FREEDOM PER SECTOR MODEL

  Fig.5. STABILITY LOOP FOR THE TUNED AND MISTUNED MODEL

Fig. 7 Figure 9

 79 Fig.7. RECONSTRUCTION SIGNAL

Figures

  Figures 10 and 11 present the results obtained by the standard discrete Fourier transform (FFT).

Fig

  Fig.12. ERRORS IN EIGENFREQUENCIES

Blade n° 1 Blade n° 77 Blade n° 1 Blade n° 77 Blade n° 1 Blade n° 77 Blade n° 1 OPR 1 OPR 2 OPR 3 OPR 4 Measures of sensor number two Blade n° 1 Blade n° 77 Blade n° 1 Blade n° 77 Blade n° 1 Blade n° 77

	Measures of the once-per-
	revolution probe (OPR)
	OPR 1	OPR 2	OPR 3	OPR 4
				Time (s)
	Measures of sensor
	number one
	OPR 1	OPR 2	OPR 3	OPR 4
				Time (s)

APPENDIX A -C, M, & K MATRICES

The mass matrix M is given by the following expression:

(32) where , m1 is equal to 4.78 kg, and m2 is equal to 30.79 kg.

The structural stiffness matrix K is given by the following expression:

where and k1 is equal to 6.7206 10

PP

(P is the matrix of conservative mode shapes), and the damping coefficient ξ is equal to 0.02 %.

The damping matrix with aeroelastic effects is given by the following relation: (36)

The stiffness matrix with aeroelastic effects is given by the following expression:

where

and ω is the angular excitation frequency.

The expression for aeroelastic coefficients is given by the following relations: The following relation can be used to compute the explicit representation of K opt (m)(t).