Jean-Luc Jany
email: jean-luc.jany@univ-brest.fr

Georges Barbier

Culture-Independent Methods for Identifying Microbial Communities in Cheese

This review focuses on the culture-independent methods available for the description of both bacterial and fungal communities in cheese. Important steps of the culture-independent strategy, which relies on bulk DNA extraction from cheese and polymerase chain reaction (PCR) amplification of selected sequences, are discussed. We critically evaluate the identification techniques already used for monitoring microbial communities in cheese, including PCR-ccdenaturing gradient gel electrophoresis (PCR-DGGE), PCR-temporal temperature gradient gel electrophoresis (PCR-TTGE) or single-strand conformation polymorphism-PCR (SSCP-PCR) as well as some other techniques that remain to be adapted to the study of cheese communities. Further, our analysis draws attention to the lack of data available on suitable DNA sequences for identifying fungal communities in cheese and proposes some potential DNA targets.

Microbial communities in cheese

Cheese is produced throughout the world and more than 1000 varieties [START_REF] Sandine | Microbially induced flavors and fermented foods: flavor in fermented dairy products[END_REF] with different forms and flavours exist. Cheesemaking is conjectured to date back to some 8000 years ago originating from the Middle East [START_REF] Fox | Fundamentals of Cheesse Science[END_REF] where the first fermented milk-based foods were made. Four basic ingredients are required to produce most cheeses: milk, rennet, salt and microorganisms. These four ingredients are processed through different steps such as acidification, coagulation, syneresis and ripening (see [START_REF] Fox | Fundamentals of Cheesse Science[END_REF].

Each unique combination of ingredients and processing parameters leads to a specific type of cheese with unique properties. [START_REF] Fox | Fundamentals of Cheesse Science[END_REF] mentioned how fascinating it is that "such a diverse range of products can be produced" from "basically similar raw material". The composition and activity of the microflora is the least controllable of all the parameters. The microflora is made up of (i) starter lactic acid bacteria that are involved in acid production during cheese manufacture and that also participate in the ripening processes to various extents; and (ii) non-starter lactic acid bacteria, other bacteria, yeasts and filamentous fungi that form the secondary microflora, which plays a significant role during ripening [START_REF] Fox | Fundamentals of Cheesse Science[END_REF]. Both starter and secondary flora modify the physical and chemical properties of cheese, contributing to and reacting to environmental changes that occur during the manufacture and ripening of cheese. [START_REF] Coppola | Dairy products[END_REF] distinguished seven technological production phases of cheese that may constitute selective pressures for the microbial species that play an important role in community succession occurring throughout cheese manufacture and ripening. The fundamental features that influence the dynamics of the cheese ecosystem have been described by [START_REF] Beresford | Recent advances in cheese microbiology[END_REF]. They include (i) physical features such as moisture, salt concentration, pH or redox potential which change during cheese manufacture and are themselves influenced by the microflora, and (ii) biological features such as those resulting from the interactions between microorganisms. The characteristics of a given cheese therefore depend on microflora dynamics. Although food microflora is undoubtedly not as diverse as environmental microflora, such as those found in the soil [START_REF] Garbeva | Microbial diversity in soil: selection of microbial populations by plant and soil type and implications for disease suppressiveness[END_REF], cheese microbial populations still remain difficult to control due to their complex dynamics and to their interactions [START_REF] Beresford | Recent advances in cheese microbiology[END_REF]. Knowledge of the structure and dynamics of the whole microbial community of cheese would promote better understanding of how cheese characteristics vary with respect to microbial growth and metabolism. For instance, greater control over microflora composition would make it possible to better select for specific organoleptic properties or to prevent quality defects or spoilage. For these reasons, cheese bacterial and fungal communities have already been partially identified using traditional methods and, to a lesser extent, molecular techniques.

Towards culture-independent methods

As in other fields in microbiology, species identification in cheese can be assessed through the use of either culture-dependent or culture-independent methods. Culture-dependent methods consist of isolating and culturing microorganisms prior to their identification according to either morphological, biochemical or genetic characteristics. Different cultures can even be bulked and analysed using global analysis methods, such as those as described below for culture-independent methods [START_REF] Ercolini | The potential of a polyphasic PCR-DGGE approach in evaluating microbial diversity of natural whey cultures for water-buffalo Mozzarella cheese production: bias of culture-dependent and cultureindependent analyses[END_REF][START_REF] Ercolini | PCR-DGGE fingerprinting: novel strategies for detection of microbes in food[END_REF]. These methods have already shed light on the structure of microbial populations during cheese manufacture [START_REF] Andrighetto | Molecular identification and cluster analysis of homofermentative thermophilic lactobacilli isolated from dairy products[END_REF][START_REF] Fitzsimmons | Phenotypic and genotypic characterization of non-starter lactic acid bacteria in mature Cheddar cheese[END_REF][START_REF] Mannu | Mesophilic lactobacilli in Fiore Sardo cheese: PCR identification and evolution during ripening[END_REF][START_REF] Berthier | Origin and diversity of mesophilic lactobacilli in Comté cheese, as revealed by PCR with repetitive and species-specific primers[END_REF][START_REF] Dasen | Genotypic and phenotypic characterization of the dynamics of the lactic acid bacterial population of adjunct-containing Cheddar cheese manufactured from raw and microfiltered pasteurized milk[END_REF]. However, culture-dependent methods are time-consuming, due to long culture periods and elaborate culture techniques. They are therefore not amenable to monitoring community dynamics during cheese manufacture and ripening. Moreover, species occurring in low numbers are often out-competed in vitro by numerically more abundant microbial species [START_REF] Hugenholtz | Impact of culture-independent studies on the emerging phylogenetic view of bacterial diversity[END_REF]) and some species may be unable to grow in vitro [START_REF] Ward | 16S rRNA sequences reveal uncultured inhabitants of a well-studied thermal community[END_REF][START_REF] Ward | Ribosomal RNA analysis of microorganisms as they occur in nature[END_REF][START_REF] Head | Microbial evolution, diversity, and ecology: a decade of ribosomal RNA analysis of uncultivated microorganisms[END_REF]. Hence, if culture conditions are poor and the number of isolates too low, the culture collection will not be representative of the community and the actual microbial diversity will be misinterpreted.

Community-level studies are relying more and more on culture-independent methods based on the direct analysis of DNA (or RNA) without any culturing step. These methods are based on protocols where total DNA (or RNA) is directly extracted from the substrate.

Coupled with a global analysis, these methods make it possible to study the total diversity from the bulk extract in a single step. As they are fast and potentially more exhaustive, these methods are well suited for analysing microbial communities over time and may provide the possibility of exploring cheese microflora dynamics in detail. Most of these methods use polymerase chain reaction (PCR) amplification of total DNA. The PCR amplicons from different species are discriminated by using gel or capillary separation or by hybridization to specific probes (Figure 1). However, these methods have potential biases, which will be discussed below.

'Pick'em all'

As previously mentioned, the isolation step of culture-dependent methods introduces biases because some species are unable to grow under the selected experimental conditions. Cultureindependent methods typically aim at collecting DNA from the whole community to overcome this bias. Nevertheless, technical issues may arise: DNA may not be recovered from all genotypes or PCR amplification may be inaccurate. Some genotypes may remain undetected due to low species abundance in the substrate, low species availability due to insufficient homogenization of the matrix, inadequate cell lysis that prevents release of nucleic acids, or inhibition of PCR amplification.

Complete homogenization of the cheese matrix can be challenging and partial homogenization may hamper cell lysis, lowering DNA availability. Therefore, existing protocols for DNA extraction from cheese include mechanical homogenization in salt solutions such as trisodium-citrate [START_REF] Coppola | Evaluation of microbial diversity during the manufacture of Fior di Latte di Agerola, a traditional raw milk pasta-filata cheese of the Naples area[END_REF][START_REF] Parayre | Easy DNA extraction and optimisation of PCR-Temporal temperature gel electrophoresis to identify the predominant high and low GC-content bacteria from dairy products[END_REF]). However, these protocols may have to be adapted to the cheese under investigation. The extraction step is usually performed in specific buffers, such as phosphate buffers containing various detergents. Extraction can also be done with guanidine thiocyanate and N-laurylsarcosine [START_REF] Callon | Application of SSCP-PCR fingerprinting to profile the yeast community in raw milk Salers cheeses[END_REF]Duthoit et al. 2005a,b;[START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF][START_REF] Bourhis | Contribution of C. beijerinckii and C. sporogenes in association to C. tyrobutyricum to the butyric fermentation in Emmental type cheese[END_REF]) and/or lytic enzymes such as pronase [START_REF] Florez | Fungal diversity and succession during the manufacture and ripening of traditional, Spanish, blue-veined Cabrales cheese, as determined by PCR-DGGE[END_REF][START_REF] El Baradei | Biodiversity of bacterial ecosystems in traditional Egyptian Domiati cheese[END_REF] or proteinase K [START_REF] Ercolini | Bacterial community structure and location in Stilton cheese[END_REF][START_REF] Cocolin | Optimization of conditions for profiling bacterial populations in food by cultureindependent methods[END_REF]). Lysis can be mechanically improved by vigorously shaking samples in tubes containing beads [START_REF] Randazzo | Diversity, dynamics, and activity of bacterial communities during production of an artisanal Sicilian cheese as evaluated by 16S rRNA analysis[END_REF][START_REF] Duthoit | Bacterial community dynamics during production of registered designation of origin Salers cheese as evaluated by 16S rRNA single-strand conformation polymorphism analysis[END_REF], Duthoit et al. 2005a,b;Feurer et al. 2004a,b;[START_REF] Delbès | Design and application of Staphylococcus-specific single strand conformation polymorphism-PCR analysis to monitor Staphylococcus populations diversity and dynamics during production of raw milk cheese[END_REF][START_REF] Florez | Fungal diversity and succession during the manufacture and ripening of traditional, Spanish, blue-veined Cabrales cheese, as determined by PCR-DGGE[END_REF][START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF][START_REF] Bourhis | Contribution of C. beijerinckii and C. sporogenes in association to C. tyrobutyricum to the butyric fermentation in Emmental type cheese[END_REF]). New commercially available automated instruments exploiting different techniques, such as pressure cycling technology, which generates alternating hydrostatic pressure [START_REF] Tao | Pressure cycling technology (PCT) applications in extraction of biomolecules from challenging biological samples[END_REF], or optimized mechanical grinding, should be tested for their capacity to improve DNA extraction from microorganisms in cheese.

Like for other food materials or environmental samples, DNA extraction yield and PCR sensitivity are significantly reduced by a wide range of inhibiting substances [START_REF] Wilson | Inhibition and facilitation of nucleic acid amplification[END_REF]. DNA extraction efficiency may be decreased by the high quantities of macromolecules in cheeses, such as casein or lipids, that can adsorb detergents, chaotropic or chelating agents necessary for nucleic acid extraction [START_REF] Bonaïti | Novel extraction strategy of ribosomal RNA and genomic DNA from cheese for PCR-based investigations[END_REF]. Moreover, even if DNA yield is high, inhibitors that have not been eliminated may lower PCR sensitivity [START_REF] Wilson | Inhibition and facilitation of nucleic acid amplification[END_REF]. For example, [START_REF] Drake | Rapid detection and identification of Lactobacillus spp. in dairy products by using the polymerase chain reaction[END_REF] reported that protein breakdown products decreased the sensitivity of PCR amplification of rRNA genes of Lactobacillus spp.. DNA solutions are therefore often purified using phenol-chloroform, which eliminates proteins and various remnants [START_REF] Randazzo | Diversity, dynamics, and activity of bacterial communities during production of an artisanal Sicilian cheese as evaluated by 16S rRNA analysis[END_REF][START_REF] Randazzo | Artisanal and experimental Pecorino Siciliano cheese: microbial dynamics during manufacture assessed by culturing and PCR-DGGE analyses[END_REF][START_REF] Cocolin | The late blowing in cheese: a new molecular approach based on PCR and DGGE to study the microbial ecology of the alteration process[END_REF]Feurer et al. 2004a,b;[START_REF] Callon | Application of SSCP-PCR fingerprinting to profile the yeast community in raw milk Salers cheeses[END_REF][START_REF] Delbès | Design and application of Staphylococcus-specific single strand conformation polymorphism-PCR analysis to monitor Staphylococcus populations diversity and dynamics during production of raw milk cheese[END_REF][START_REF] Duthoit | Bacterial community dynamics during production of registered designation of origin Salers cheese as evaluated by 16S rRNA single-strand conformation polymorphism analysis[END_REF]Duthoit et al. , 2005a,b;,b;[START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF][START_REF] Bourhis | Contribution of C. beijerinckii and C. sporogenes in association to C. tyrobutyricum to the butyric fermentation in Emmental type cheese[END_REF]). Several authors have used commercial DNA extraction kits based on ion-exchange chromatography, size-exclusion or sorption techniques [START_REF] Ercolini | The potential of a polyphasic PCR-DGGE approach in evaluating microbial diversity of natural whey cultures for water-buffalo Mozzarella cheese production: bias of culture-dependent and cultureindependent analyses[END_REF][START_REF] Rademaker | The surface microflora dynamics of bacterial smear-ripened Tilsit cheese determined by T-RFLP DNA population fingerprint analysis[END_REF] or just for DNA purification [START_REF] Parayre | Easy DNA extraction and optimisation of PCR-Temporal temperature gel electrophoresis to identify the predominant high and low GC-content bacteria from dairy products[END_REF].

It is noteworthy that most studies optimized DNA extraction to target bacterial DNA, with the exception of [START_REF] Callon | Application of SSCP-PCR fingerprinting to profile the yeast community in raw milk Salers cheeses[END_REF], who evaluated the dynamics of yeasts, and [START_REF] Florez | Fungal diversity and succession during the manufacture and ripening of traditional, Spanish, blue-veined Cabrales cheese, as determined by PCR-DGGE[END_REF], who investigated the dynamics of the dominant microorganisms including bacteria, yeasts and filamentous fungi. In order to characterise whole microbial communities, protocols have to be adapted to extract DNA from all different types of microorganisms, as attempted by [START_REF] Bonaïti | Novel extraction strategy of ribosomal RNA and genomic DNA from cheese for PCR-based investigations[END_REF].

Most culture-independent methods rely on PCR amplification of a targeted sequence. This step can be affected by preferential or differential PCR amplification [START_REF] Reysenbach | Differential amplification of rRNA genes by polymerase chain reaction[END_REF][START_REF] Walsh | Preferential PCR amplification of alleles: mechanisms and solutions[END_REF]Ercolini 2003;Kanawaga 2003) that may hinder the detection of some genotypes when analysing bulk DNA extracted from cheese. Preferential PCR amplification can be caused by (i) primer mismatches at the annealing sites of the DNA templates of some genotypes or (ii) a lower rate of primer hybridization to certain templates due to differential denaturation of these templates [START_REF] Walsh | Preferential PCR amplification of alleles: mechanisms and solutions[END_REF][START_REF] Suzuki | Bias caused by template annealing in the amplification of mixtures of 16S rRNA genes by PCR[END_REF]. Systematically testing different sets of primers and enhancing DNA denaturation during PCR by using different reagents (denaturants and cosolvents) may solve these problems [START_REF] Reysenbach | Differential amplification of rRNA genes by polymerase chain reaction[END_REF][START_REF] Weissensteiner | Strategy for controlling preferential amplification and avoiding false negatives in PCR typing[END_REF][START_REF] Hansen | Biased 16S rDNA PCR amplification caused by interference from DNA flanking the template region[END_REF].

A second type of bias that may affect PCR carried out on complex bulk DNA extracts is the occurrence of (i) heteroduplexes that arise in later PCR cycles when primer concentration decreases and the concentration of PCR products is high enough to compete with the primers for annealing (Kanawaga 2003) and (ii) chimeric amplicons that also form in later PCR cycles when the concentration of incompletely extended primers is high enough to compete with the original primer for annealing, or when template concentration is high enough to allow the re-annealing of templates before or during primer extension (Kanawaga 2003). All these artefacts can generate additional signals that do not correspond to genotypes in the sample. These artefacts can be minimized by using a low number of PCR cycles.

Analysing bulk DNA

Through the construction of clone libraries from PCR products amplified from bulk cheese DNA and subsequent sequencing of the different clones, it is theoretically possible to examine the actual diversity of a given community. Although the cost of cloning and sequencing is decreasing, this strategy remains expensive if used on a routine basis since it implies large-scale cloning and sequencing to ensure that the community is exhaustively screened. This strategy has been used for the study of bacterial cheese communities (Feurer et al. 2004a,b;[START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF][START_REF] El Baradei | Biodiversity of bacterial ecosystems in traditional Egyptian Domiati cheese[END_REF]), but only as a complement to other techniques (Table 1). Amongst the other molecular methods that may allow exhaustive screening of microbial communities with no requirement for cloning PCR products, the most commonly used in cheese microbiology are based on gel electrophoresis or chromatography.

The principal techniques that can be used to describe microbial communities in cheese are briefly discussed below.

PCR-denaturing gradient gel electrophoresis and PCR-temporal temperature gradient gel electrophoresis

PCR-denaturing gradient gel electrophoresis (PCR-DGGE; Myers 1987) and PCR-temporal temperature gradient gel electrophoresis (PCR-TTGE; [START_REF] Yoshino | Temperature sweep gel electrophoresis: a simple method to detect point mutations[END_REF]) (collectively referred to here as PCR-DG/TTGE) are based on the separation of PCR amplicons of the same size but with different sequences. In a denaturing acrylamide gel, DNA partially denatures in discrete regions called melting domains. The melting temperature of these domains is sequence specific. When the melting temperature (Tm) of the lowest melting domain is reached, the DNA is partially denatured creating branched molecules. This branching reduces DNA mobility in the gel. Therefore, amplicons of the same size but with different nucleotide compositions can be separated based on differences in the melting behaviour of their melting domains. For PCR-DGGE, the denaturing conditions rely on the use of chemical denaturants (formamide and urea) incorporated into an acrylamide gel as a linear denaturing gradient. PCR-DGGE electrophoresis is carried out at constant temperature, typically between 55°C and 65°C [START_REF] Ercolini | PCR-DGGE fingerprinting: novel strategies for detection of microbes in food[END_REF]. For PCR-TTGE, the denaturing gradient is obtained by varying the temperature over time without chemicals, thus generating more reproducible data. PCR-DG/TTGE provides optimal resolution when PCR products are not completely denatured. Thus, adding a so-called GC clamp (30-40bp) to one of the PCR primers ensures that the PCR products are not completely denatured during the analysis (Myers et al. 1985a,b;Shefield et al. 1989). Prior to PCR-DG/TTGE analysis, the location of the different melting domains within a DNA sequence can be predicted in silico, allowing selection of the best primer pair. A PCR-DG/TTGE database containing migration profiles corresponding to reference strains is generally created [START_REF] Ogier | Identification of the bacterial microflora in dairy products by temporal temperature gradient gelelectrophoresis[END_REF][START_REF] Ogier | Molecular fingerprinting of dairy microbial ecosystems by use of temporal temperature and denaturing gradient gel electrophoresis[END_REF] to facilitate further identification of PCR-DG/TTGE profiles. A molecular ladder can be constructed by using amplicons corresponding to representative species of this database and used in each gel migration. However, this kind of database cannot be exhaustive and representative of the actual community analysed and usually requires constant updates. The advantage of PCR-DG/TTGE is that amplicons can be directly extracted from the DG/TTGE acrylamide gel and sequenced. Unidentified profiles that do not match reference profiles can thus be sequenced, compared to publicly available sequence databases and appended to the PCR-DG/TTGE profile database. The first PCR-DGGE application to microbiology was done by [START_REF] Muyzer | Phylogenetic relationships of Thiomicrospora species and their identification in deep-sea hydrothermal vent samples by denaturing gradient gel electrophoresis of 16S rDNA fragments[END_REF] who studied bacterial communities of deep-sea hydrothermal vents. PCR-DG/TTGE are the molecular techniques that have been the most extensively used for the study of microbial communities in dairy products (e.g. [START_REF] Lafarge | Le potentiel de la TTGE pour l'étude bactérienne de quelques laits crus[END_REF][START_REF] Ogier | Identification of the bacterial microflora in dairy products by temporal temperature gradient gelelectrophoresis[END_REF][START_REF] Ercolini | PCR-DGGE fingerprinting: novel strategies for detection of microbes in food[END_REF] and more specifically to monitor the structure and even dynamics of microbial communities in cheese (see references in Table 1).

Single-strand conformation polymorphism-PCR

Single-strand conformation polymorphism-PCR (SSCP-PCR; [START_REF] Orita | Detection of polymorphisms of human DNA by gel electrophoresis as single-strand conformation polymorphisms[END_REF][START_REF] Ravnik-Glavac | Sensitivity of single-strand conformation polymorphism and heteroduplex method for mutation detection on the cystic fibrosis gene[END_REF]) is a technique using either acrylamide gel-or capillary-based automated sequencer, based on the separation of denatured (single-stranded) PCR products. Under nondenaturing conditions, single-stranded DNA folds into tertiary structures according to their nucleotide sequences and their physicochemical environment (e.g., temperature and ion strength). This causes differences in electrophoretic mobility in non-denaturing gels. SSCP-PCR is potentially easier to carry out than PCR-DG/TTGE since there is no need for gradient gels or use of GC-clamp primers and it can be performed using an automated sequencer.

However, when using an automated sequencer, one of the disadvantages of this technique lies in the difficulty of appending new data to an existing database: samples presenting unknown profiles cannot be directly sequenced because they are labelled. SSCP-PCR is the second most-used method to study microbial communities of cheese (see references in Table 1).

Terminal restriction fragment length polymorphism

Terminal restriction fragment length polymorphism (T-RFLP; [START_REF] Liu | Characterization of microbial diversity by determining terminal restriction fragment length polymorphisms of genes encoding 16S rRNA[END_REF]) is based on digestion of fluorescent end-labelled PCR products with restriction endonucleases. Either one or both 5'and 3' ends of the amplicon can be labelled by incorporating a dye on either one or both PCR primers. The digested products are separated by electrophoresis using either acrylamide gel-or capillary-based automated sequencer, with laser detection of the labelled fragments. This system only detects the end-labelled terminal restriction fragments (TRFs) of the digested PCR products and their size can be calculated based on the use of DNA size standards that are run simultaneously with the samples. The data consists of the sizes of the PCR amplicons that contain the labelled primer and are observed as electrophoregram peaks or gel bands. Variation in the presence and location of the restriction sites results in different genotypes having different TRF lengths. T-RFLP was initially developed as a fingerprinting technique [START_REF] Liu | Characterization of microbial diversity by determining terminal restriction fragment length polymorphisms of genes encoding 16S rRNA[END_REF], with the number of TRFs used as an indication of biodiversity.

The use of a database of TRF profiles obtained from reference samples allows the identification of the different species of a given community [START_REF] Dickie | Vertical niche differentiation of ectomycorrhizal hyphae in soil as shown by T-RFLP analysis[END_REF]. As for SSCP-PCR performed using an automated sequencer, samples that present unknown profiles cannot be directly sequenced because they are cut and labelled. T-RFLP has been used to study diverse microbial communities (e.g. [START_REF] Liu | Determination of the microbial diversity of anaerobic-aerobic activated sludge by a novel molecular biological technique[END_REF] and has been extensively used by mycologists since this method is reportedly more sensitive than PCR-DG/TGGE for fungi [START_REF] Brodie | Soil fungal community structure in a temperate upland grassland soil[END_REF]. However, some limitations-due in particular to inefficient restriction enzyme cleavage-have been pointed out [START_REF] Avis | A 'dirty' business: testing the limitations of terminal restriction fragment length polymorphism (TRFLP) analysis of soil fungi[END_REF]. T-RFLP was also found to be an excellent tool for culture-independent assessment of bacterial community structure and dynamics during ripening of cheese [START_REF] Rademaker | The surface microflora dynamics of bacterial smear-ripened Tilsit cheese determined by T-RFLP DNA population fingerprint analysis[END_REF].

Denaturing high-performance liquid chromatography

Denaturing high-performance liquid chromatography (DHPLC) (Oefner andUnderhill 1995 in Xiao and[START_REF] Xiao | Denaturing high-performance liquid chromatography: A review[END_REF]) allows separation of amplicons using an ion-pair reversed-phase high-performance liquid chromatography (IP RP HPLC) automated detection system. It was initially used to detect single nucleotide polymorphisms (SNPs) in clinical applications (e.g., [START_REF] Frueh | Use of denaturing high-performance liquid chromatography for rapid detection and identification of seven Candida species[END_REF] based on the analysis of heteroduplex formation. DHPLC is a promising approach for microbial community analysis [START_REF] Barlaan | Profiling and monitoring of microbial populations by denaturing high-performance liquid chromatography[END_REF]. PCR amplicons are injected into a chromatography column containing alkylated non-porous polystyrene/polydivinylbenzene particles. Separation of the different amplicons relies on the elution of partially denatured PCR products, which is achieved with the intervention of the ion-pairing agent, triethylammonium acetate (TEAA) and the cartridge matrix of the system.

As described in [START_REF] Barlaan | Profiling and monitoring of microbial populations by denaturing high-performance liquid chromatography[END_REF], under partial denaturation and a given flow rate of gradient buffers, amplicons of the same size but with different melting behaviours due to different nucleotide compositions will have different retention times. These differences are due to the reduced negative charges in the single-stranded portions of the partially denatured PCR products compared to the double-stranded molecules. The negative charges interact with the positive charges of ammonium ions of TEAA, and double-stranded DNA is more efficiently adsorbed to the stationary phase in the cartridge. DHPLC permits high-throughput automated analyses and, unlike SSCP-PCR or T-RFLP, it allows the collection of elution fractions corresponding to different amplicons that can be directly sequenced even more easily than with PCR-DG/TTGE methods. Protocols have already been developed for analysing marine bacterial populations [START_REF] Barlaan | Profiling and monitoring of microbial populations by denaturing high-performance liquid chromatography[END_REF], monitoring intestinal microflora [START_REF] Goldenberg | Molecular monitoring of the intestinal flora by denaturing high performance liquid chromatography[END_REF]) and, recently, also for studying the bacterial diversity occurring in natural whey cultures used for cheese manufacture [START_REF] Ercolini | Microbial diversity in natural whey cultures used for the production of Caciocavallo Silano PDO cheese[END_REF]. The latter study showed that DHPLC technique was at least as effective as the widely used PCR-DGGE technique in assessing species diversity of food-related microbial communities.

DNA microarrays

The complementary DNA (cDNA) microarray (or microchips) technology has dramatically changed the way gene expression can be assessed [START_REF] Duggan | Expression profiling using cDNA microarrays[END_REF]. Since [START_REF] Guschin | Oligonucleotide microchips as genosensors for determinative and environmental studies in microbiology[END_REF] introduced the DNA microarray approach to microbial community analysis, the DNA array-based methods hold great promise for more extensive analyses of microbial communities [START_REF] Zhou | Microarrays for bacterial detection and microbial community analysis[END_REF]Bodrossy et al. 2007;[START_REF] Wagner | Unravelling microbial communities with DNA-microarrays: challenges and future directions[END_REF]) and potential applications cover various fields of microbiology, including food science [START_REF] Bae | Development and evaluation of genome-probing microarrays for monitoring lactic acid bacteria[END_REF].

By using DNA microarrays, the identification of labelled PCR products or directly retrieved RNA relies on their hybridization to oligonucleotide probes attached to a substrate. A description of the various types of microarrays is given by [START_REF] Zhou | Microarrays for bacterial detection and microbial community analysis[END_REF]. Unlike the previously described methods, DNA array technology potentially allows the simultaneous application of a nearly unlimited number of probes in a single hybridization experiment [START_REF] Small | Direct detection of 16S rRNA in soil extracts by using oligonucleotide microarrays[END_REF][START_REF] Peplies | Optimization strategies for DNA microarraysbased detection of bacteria with 16S rRNA-targeting oligonucleotide probes[END_REF]. This technique is therefore very well suited to even the most complex environmental samples. However, for this approach to work, each probe must specifically hybridize, under given stringency conditions, to a fully matched DNA target [START_REF] Valinsky | Analysis of bacterial community composition by oligonucleotide fingerprinting of rRNA genes[END_REF] which has been proven very difficult [START_REF] Wagner | Unravelling microbial communities with DNA-microarrays: challenges and future directions[END_REF].

Moreover, the design and refinement of efficient probes depend on the comprehensiveness and quality of probe target database [START_REF] Wagner | Unravelling microbial communities with DNA-microarrays: challenges and future directions[END_REF]. The low quality of some annotated sequences in the available databases complicates probe design. Nonetheless, given the lower diversity of food-borne communities (compared to environmental communities) and given the unequalled high-throughput capacity of DNA microarrays, it would be worthwhile to devote research efforts for developing a robust probe target sequence database necessary for using DNA microarrays as a routine tool for monitoring cheese microbial communities throughout cheese manufacture.

What is a good target?

The discrimination between the different species from a cheese sample depends on the capacity of the analytical technique to reveal polymorphisms. However, whatever analytical technique is used, no discrimination is possible without polymorphism. The target DNA sequence must therefore be variable between species. In addition, since almost all of the above-described techniques rely on PCR amplification, the target DNA sequence must also include conserved sequences that can serve as anchors for PCR primers.

Unlike prokaryotic genes that do not usually harbour introns, fungal genes are interrupted by non-coding introns. Although introns are much more variable than exons and thus offer high levels of polymorphism between species, their length and their location vary greatly beyond the family level [START_REF] James | Reconstructing the early evolution of Fungi using a sixgene phylogeny[END_REF]. It is therefore difficult to design primers that allow PCR amplification of introns from all the species in the community. Moreover, the analysis of exons is often discriminative enough to distinguish between species since they show sufficient variation at the interspecific level, especially at their third codon positions that evolve at rates similar to introns. Consequently, most of the DNA regions selected for studying fungal communities are located in exons.

Bacterial and fungal ribosomal DNA

Since Woese and Fox's paper (1977), most community surveys have focused on RNA genes and intergenic spacers. Bacterial 16S, 23S and 5S rRNA genes are organized into a cotranscribed operon. From one to as many as 15 copies of the operon may be dispersed in a bacterial genome [START_REF] Klappenbach | rRNA operon copy number reflects ecological strategies of bacteria[END_REF]. In contrast, in fungi and more generally in eukaryotes, 18S, 28S 5.8S and 5S rRNA genes and intergenic spacers are tandemly repeated (up to hundreds of repeats) [START_REF] Cihlar | The organization of the ribosomal RNA genes in the fungus Mucor racemosus[END_REF]. The 5S rRNA is, in some species, separated from the main unit (Kellog and Appels 1995). These repeated units are organized as multiple and dispersed arrays at chromosomal nucleolar organizer regions (NORs). In both prokaryotes and eukaryotes, rRNA genes usually show extraordinary sequence homogeneity within a species [START_REF] Elder | Concerted evolution of repetitive DNA sequences in Eukaryoteseukaryotes[END_REF][START_REF] Liao | Concerted evolution: molecular mechanism and biological implications[END_REF]. This sequence homogenization is likely to be due to a process known as concerted evolution. The underlying mechanisms of concerted evolution involve recombination processes collectively referred to as mitotic or somatic recombination (see [START_REF] Liao | Gene conversion drives within genic sequences: concerted evolution of ribosomal RNA genes in bacteria and archaea[END_REF] including unequal crossing-over and gene conversion that are responsible for DNA repair during replication and transcription. The paradigm of concerted evolution makes it possible to treat repeated rRNA genes as one locus. For this reason, rRNA genes have been widely used in phylogeny reconstructions and species identification. However, deviations from this expected homogeneity of rDNA sequences have been detected in many taxa.

Although multiple-gene investigation has already been recommended for bacterial species identification [START_REF] Stackebrandt | Report of the ad hoc committee for the re-evaluation of the species definition in bacteriology[END_REF], studies evaluating bacterial diversity and/or dynamics or investigating bacterial phylogenetics have mostly focused only on ribosomal DNA array analysis [START_REF] Throbäck | Reassessing PCR primers targeting nirS, nirK and nosZ genes for community surveys of denitrifying bacteria with DGGE[END_REF]). In cheese, all bacterial community surveys are based on the analysis of 16S rRNA genes and 16S-23S intergenic region (Table 1). Bacterial 16S rRNA genes comprise nine hypervariable regions, V1-V9, that exhibit considerable sequence diversity among species [START_REF] Van De Peer | A quantitative map of nucleotide substitution rates in bacterial rRNA[END_REF][START_REF] Baker | Review and re-analysis of domain-specifc 16S primers[END_REF]. These hypervariable regions are generally flanked by conserved sequences that can serve as anchors for universal or specific primer pairs (see [START_REF] Baker | Review and re-analysis of domain-specifc 16S primers[END_REF]. They are therefore used for species identification and allow the evaluation of community diversity. No single region can differentiate among all bacteria and different regions can be used depending on the goal [START_REF] Chakravorty | A detailed analysis of 16S ribosomal RNA gene segments for the diagnosis of pathogenic bacteria[END_REF]). In cheese, the V3 hypervariable region is the target that has been the most extensively used, but the use of different sets of primers targeting different regions can improve the analysis [START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF].

In fungi, the analysis of fungal rRNA genes limits identification to the genus or family level [START_REF] Anderson | Diversity and ecology of soil fungal communities: increased understanding through the application of molecular techniques[END_REF]. The fungal internal transcribed spacers (ITS) provide a greater taxonomic resolution than rRNA genes and are generally used for fungal community surveys in different environments [START_REF] Anderson | Diversity and ecology of soil fungal communities: increased understanding through the application of molecular techniques[END_REF]. The ITS is a region located between the 18S rRNA and 28S rRNA genes and including the 5.8S rRNA gene that splits the ITS into two parts: ITS1 and ITS2. The ITS region undergoes a faster rate of evolution than rRNA genes but its sequence remains homogenous within a species. Indeed, both ITS1 and ITS2 fulfil significant functions during rRNA maturation (Joseph et al. 1991;[START_REF] Liu | A conserved sequence in internal transcribed spacer 1 of plant nuclear genes[END_REF] and are under selective pressure. In Penicillium spp. which are amongst the most prominent fungi in cheese, the analysis of ITS affords better discrimination than rRNA genes [START_REF] Skouboe | Molecular methods for differentiation of closely related Penicillium species[END_REF][START_REF] Doaré-Lebrun | Analysis of fungal diversity of grapes by application of temporal temperature gradient gel electrophoresis -potentialities and limits of the method[END_REF]). Nevertheless, [START_REF] Doaré-Lebrun | Analysis of fungal diversity of grapes by application of temporal temperature gradient gel electrophoresis -potentialities and limits of the method[END_REF] when analysing the fungal community in grape showed that the use of ITS was not sufficient for good interspecific discrimination, especially for the species of the subgenus Penicillium, a monophyletic group of moulds that represent 58 of the ≈250 accepted species in the genus Penicillium [START_REF] Seifert | Prospects for fungus identification using COI DNA barcodes, with Penicillium as a test case[END_REF]. It is therefore recommended to use other targets in addition to ITS when analysing fungal communities in cheese where species of this subgenus are among the most prominent.

Other fungal genes

With the exception of rRNA genes and ITS, only a few number of sequences have been used for studying fungal communities. Recently, [START_REF] Seifert | Prospects for fungus identification using COI DNA barcodes, with Penicillium as a test case[END_REF] tested the reliability of using the CO1 gene that codes for mitochondrial cytochrome c oxidase 1 as a barcoding tool for Penicillium spp. identification. This gene has already been used extensively in animal barcoding and could be used as a new marker to investigate fungal communities. Sequence analysis of CO1 yielded a coherent phylogeny of the taxonomically challenging Penicillium subgenus [START_REF] Seifert | Prospects for fungus identification using COI DNA barcodes, with Penicillium as a test case[END_REF], suggesting that CO1 is a powerful tool for fungal barcoding.

However, the results by these authors show that CO1 is not more variable among the species of the subgenus Penicillium than ITS. The low CO1 interspecific divergence precludes distinguishing species of the subgenus Penicillium using gel/capillary separation of PCR products. Amongst other possible targets, genes that encode mitochondrial rRNA genes [START_REF] Lutzoni | Assembling the fungal tree of life: progress, classification and evolution of subcellular traits[END_REF]) could represent an alternative to nuclear rRNA genes. Moreover, the general effort for inferring phylogeny of the kingdom fungi [START_REF] Lutzoni | Assembling the fungal tree of life: progress, classification and evolution of subcellular traits[END_REF][START_REF] Blackwell | Research coordination networks: a phylogeny for kingdom fungi (Deep Hypha)[END_REF][START_REF] Hibbet | A higher-level phylogenetic classification of the Fungi[END_REF]) has yielded a high number of sequences for different genes and for different fungal species that can be tested for species identification. The non-ribosomal genes used for inferring fungal phylogenies comprise: (i) the genes RPB1 and RPB2 that encode the two largest subunits of the RNA polymerase II [START_REF] James | Reconstructing the early evolution of Fungi using a sixgene phylogeny[END_REF]; the gene EF-1 α coding for elongation factor 1-alpha [START_REF] James | Reconstructing the early evolution of Fungi using a sixgene phylogeny[END_REF]; the gene BenA encoding β-tubulin A [START_REF] Einax | Oligonucleotide primers for the universal amplification of betatubulin genes facilitate phylogenetic analyses in the regnum Fungi[END_REF][START_REF] Samson | Phylogenetic analysis of Penicillium subgenus Penicillium using partial β-tubulin sequences[END_REF]); the GPD gene coding for glyceraldehyde 3-phosphate dehydrogenase [START_REF] Berbee | Cochliobolus phylogenetics and the origin of known, highly virulent pathogens, inferred from ITS and glyceraldehyde-3-phosphate dehydrogenase gene sequences[END_REF]. All these potential targets should be tested for their usefulness for distinguishing between different species of fungi that compose fungal communities in cheese. According to [START_REF] Samson | Phylogenetic analysis of Penicillium subgenus Penicillium using partial β-tubulin sequences[END_REF] and [START_REF] Seifert | Prospects for fungus identification using COI DNA barcodes, with Penicillium as a test case[END_REF] BenA is more variable than ITS amongst the species of the Penicillium subgenus and could be a good target for analysing cheese fungal communities using methods based on gel/capillary analysis of PCR products. The most promising sequences to analyse can be identified by in silico data mining of publicly available databases. However, sequence availability for a large range of species varies amongst genes. Additional sequencing is likely to be required to cover a reasonable array of species. This requires selecting and designing universal or even a set of different primers to sequence the targeted genes in the different species investigated.

Limitations and pitfalls of culture-independent methods

As previously mentioned, the first limitation arises from the difficulty of accessing every genotype from the community: poor DNA extraction yield, PCR inhibition by various extraction by-products or by substances coming from the cheese matrix itself, and differential PCR amplification have already been reviewed [START_REF] Wintzingerode | Determination of microbial diversity in environmental samples: pitfalls of PCR-based analysis[END_REF][START_REF] Ercolini | PCR-DGGE fingerprinting: novel strategies for detection of microbes in food[END_REF].

We have already made some suggestions to address, when possible, these concerns. The various techniques have limitations in terms of resolution: PCR-DG/TTGE, SSCP-PCR, T-RFLP, DHPLC and even microarrays can generate patterns in which different genotypes group together due to co-migration/co-elution [START_REF] Ogier | Identification of the bacterial microflora in dairy products by temporal temperature gradient gelelectrophoresis[END_REF]Feurer et al. 2004a;[START_REF] Lafarge | Le potentiel de la TTGE pour l'étude bactérienne de quelques laits crus[END_REF][START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF][START_REF] El Baradei | Biodiversity of bacterial ecosystems in traditional Egyptian Domiati cheese[END_REF] or 'co-hybridization' [START_REF] Wagner | Unravelling microbial communities with DNA-microarrays: challenges and future directions[END_REF]). Since they allow direct (cloning) sequencing of migrants/eluants, PCR-DG/TTGE and DHPLC may overcome the co-elution/co-migration problem more easily. Another limitation of gel/capillary migration-based methods is obtaining profiles in which the less-common amplified sequences cannot be distinguished from background noise (Feurer et al. 2004a;[START_REF] Callon | Application of SSCP-PCR fingerprinting to profile the yeast community in raw milk Salers cheeses[END_REF]. This problem increases with the diversity of the community. Ideally, multiple DNA targets should be used, as suggested by [START_REF] Doaré-Lebrun | Analysis of fungal diversity of grapes by application of temporal temperature gradient gel electrophoresis -potentialities and limits of the method[END_REF][START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF], to increase the efficiency in discriminating between species. Another remark made by several authors [START_REF] Ercolini | The potential of a polyphasic PCR-DGGE approach in evaluating microbial diversity of natural whey cultures for water-buffalo Mozzarella cheese production: bias of culture-dependent and cultureindependent analyses[END_REF]Feurer et al. 2004a,b;[START_REF] Florez | Fungal diversity and succession during the manufacture and ripening of traditional, Spanish, blue-veined Cabrales cheese, as determined by PCR-DGGE[END_REF][START_REF] Delbès | Monitoring bacterial communities in raw milk and cheese by culture-dependent and -independent 16S rRNA gene-based analyses[END_REF]) is that culture-independent methods regularly fail to identify species obtained using culture-dependent methods. These two types of methods reveal different images of the same community. Therefore, those authors suggest that using a polyphasic approach, combining culture-dependent and culture-independent methods, may be worthwhile to obtain a more accurate view of the structure of the microbial community.

Nevertheless, culture-independent methods have proven to be the only ones with the capacity for monitoring the rapid dynamics of microbial communities during cheese manufacture and ripening processes, where microorganisms encounter multiple environmental shifts. Cultureindependent methods need to be improved to reveal as accurately as possible the actual microbial communities.

Microbial community activity in cheese

In order to reveal metabolically active populations, some authors performed analysis on reverse-transcribed RNA. [START_REF] Randazzo | Diversity, dynamics, and activity of bacterial communities during production of an artisanal Sicilian cheese as evaluated by 16S rRNA analysis[END_REF] and Duthoit et al. (2005a)

Perspectives in the field of cheese microbiology

Our review draws attention to the fact that only a few studies [START_REF] Florez | Fungal diversity and succession during the manufacture and ripening of traditional, Spanish, blue-veined Cabrales cheese, as determined by PCR-DGGE[END_REF][START_REF] Callon | Application of SSCP-PCR fingerprinting to profile the yeast community in raw milk Salers cheeses[END_REF][START_REF] Gente | Development of primers for detecting dominant yeasts in smear-ripened cheeses[END_REF]) have targeted fungal communities of cheeses. However, yeasts and filamentous fungi play a vital role in the development of organoleptic characteristics and/or, on the contrary, may be the source of quality defects. Protocols need to be developed to optimize fungal DNA extraction from cheese and new target sequences, in addition to ribosomal loci, should be investigated to better discriminate between the different fungal species found in cheese. Surveys could then be extended to the whole microbial community.

It is noteworthy that, at present, some species can only be detected by culture and that polyphasic methods, including both culture-independent and culture-dependent approaches, are necessary to at least allow evaluation of the efficiency of these two types of approaches.

Nevertheless, although culture-independent methods still fail to exhaustively describe the microbial community composition in cheese, these methods provide a much faster assessment of community composition than culture-dependent methods do. New light has been shed on population dynamics and culture-independent methods offer a powerful tool for controlling cheese manufacture. Therefore, it is worthwhile to devote efforts to improve the resolution of culture-independent methods and facilitate their transfer to cheese industry.

Because cheese microbial communities are much less complex than environmental communities (in terms of the number of species encountered), they offer an ideal system to test identification methods that require setting up reference databases or designing individual nucleic probes, such as for DNA microarrays. A major task is to select the best targets allowing both universal screening of the microorganisms in cheese and discrimination between taxa at the species level. Achieving this goal requires increasing the accessibility of high-throughput DNA sequencing technology that can provide digital images of cheese ecosystems (cheese ecogenome) by identifying genes associated with species and even functions.

  were among the first to use reverse transcriptase-PCR-DGGE (RT-DGGE) and reverse transcriptase-PCR-SSCP (RT-PCR-SSCP), respectively. By combining RT-PCR-DGGE and PCR-DGGE or RT-PCR-SSCP and PCR-SSCP, these authors were able to differentiate the active component (rRNA-derived) from the total diversity (rDNA-derived) of the community. Duthoit et al. (2005b) even attempted to link population dynamics and activities (as assessed by PCR-SSCP and RT-PCR-SSCP) to sensorial characteristics in order to identify which species play a major role in the development of the organoleptic properties of Salers cheese. However, their analysis failed to explain the diversity of sensorial qualities in Salers cheese.Nonetheless, combining DNA-and RNA-based analyses with more quantitative methods, such as DNA-and cDNA-microarrays coupled with chemiometric and/or sensorial tests, could significantly increase our ability to identify the impact of the microbial population on organoleptic characteristics.

Acknowledgements

This work was financially supported by Bretagne Biotechnologies Alimentaires and Région Bretagne (FongiLait grant). The authors would like to thank the managing committee of Bretagne Biotechnologies Alimentaires for critical reading of the manuscript and G. Le Dréan, V. Vasseur and M.A. Lebras-Quéré for stimulating discussions.


Denis Laxalde

Mathias Legrand

Nonlinear Modal Analysis of Mechanical Systems with Frictionless Contact Interfaces

Keywords: nonlinear modes, unilateral contact, Rayleigh quotient, Fourier methods, augmented Lagrangian, blades

This paper explores mechanical systems undergoing unilateral frictionless contact conditions in the framework of nonlinear modal analysis. The nonlinear eigenproblem is formulated in the frequency domain through the minimization of a Rayleigh quotient subject to non-penetration inequality constraints. An additional equality constraint is introduced for normalization purposes. The resulting constrained minimization problem is then solved using an augmented Lagrangian strategy. Two applications are proposed: a thin longitudinal rod in unilateral contact with a rigid obstacle and a turbomachinery compressor blade in contact with a rigid casing. The first application illustrates the complexity of the nonlinear modal characterization of a system experiencing unilateral contact conditions while the second demonstrates the applicability of the proposed approach to large-scale mechanical systems involving non-smooth nonlinear terms.

Introduction

The concept of nonlinear mode is now commonly accepted as a reliable and well-suited approach for the analysis of nonlinear dynamical systems [START_REF] Vakakis | Normal Modes and Localization in Nonlinear Systems[END_REF] since it offers the capability to extract the essential signature of nonlinear dynamical systems. While various analytical methods developed during the past decades have contributed to strengthen the relevant theoretical background, new challenges arise for large-scale mechanical systems [START_REF] Kerschen | Nonlinear normal modes, Part I: A useful framework for the structural dynamicist[END_REF]. As evidenced in the literature, promising techniques such as invariant manifold approaches [START_REF] Pesheck | A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds[END_REF], asymptotic-numerical methods [START_REF] Arquier | Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes[END_REF], shooting techniques [START_REF] Peeters | Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques[END_REF] or Fourier strategies [START_REF] Laxalde | Complex non-linear modal analysis for mechanical systems: Application to turbomachinery bladings with friction interfaces[END_REF] provide an appealing framework for the development of dedicated numerical tools which should bridge the gap between academic research and industrial applications.

Non smooth nonlinearities such as contact yield other difficult challenges both on numerical and phenomenological sides that are not well tackled yet. Since unilateral contact is defined by a multi-valued evolution law which takes the form of inequality constraints, solution methods often fall in the theoretical field of optimization [7,8] where the subsequent Lagrange multipliers play the role of the contact pressure field. Among other, augmented Lagrangian functionals stand as sophisticated and robust approaches to numerically approximate the latter, by transforming it into a smooth saddle point problem, and have been fruitfully applied in mechanical engineering [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF][START_REF] Laursen | A continuum-based finite element formulation for the implicit solution of multibody, large deformation-frictional contact problems[END_REF][START_REF] Pietrzak | Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment[END_REF].

Numerical methods for contact dynamics analysis are mostly based on time integration and alternatives are unfortunately rarely explored [START_REF] Woo | Application of the harmonic balance method to ground moling machines operating in periodic regimes[END_REF][START_REF] Leib | Experimental and Numerical Analysis of Two Vibro-Impacting Beams[END_REF]. Accordingly, it seems challenging to regard this class of systems in the light of nonlinear modal analysis. In the present paper, the target application concerns rotating components in turbomachinery undergoing intermittent contacts (or impacts) with surrounding stationary structures. As a matter of fact, current design trends (more flexible and thinner structures) together with operating clearance reductions for aerodynamic efficiency purposes lead to complex and poorly known phenomena [START_REF] Azeez | Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts[END_REF][START_REF] Lesaffre | Stability analysis of rotating beams rubbing on an elastic circular structure[END_REF][START_REF] Legrand | Two-dimensional modeling of an aircraft engine structural bladed disk-casing modal interaction[END_REF]. This, in particular, motivates the use of such modern techniques to extend existing design methodologies which currently mostly rely on linear analyses or time-stepping approaches.

The proposed approach is first described in the usual formalism of continuum mechanics. Eigensolutions are sought in the form of Fourier series which results in a mixed frequency/time boundary value problem. Then, introducing a variational formulation, the eigenvalue problem reduces to finding critical values of a generalized Rayleigh quotient functional in the admissible domain defined by the contact boundary conditions. This constrained minimization problem is then completed by an additional equality constraints which normalizes eigensolutions with reference with an energy. Finally, the resulting problem is solved in the framework of an augmented Lagrangian approach. Numerical approximations and algorithms are also detailed.

Two illustrative examples are investigated. The first considers a thin rod in unilateral contact with a rigid foundation and the second deals with a turbomachinery compressor blade in intermittent contact at its tip edge with a rigid casing. Consequences of unilateral contact on modal quantities are investigated.

Boundary value nonlinear eigenproblem

This section concerns the derivation of the strong nonlinear eigenvalue formulation of a flexible structure possibly in contact with a rigid foundation. Nonlinear modes [START_REF] Vakakis | Normal Modes and Localization in Nonlinear Systems[END_REF] are defined as non-trivial periodic solutions of an autonomous dynamical system. Furthermore, linear damping as well as friction are not considered so that the system is conservative. In the framework of infinitesimal deformations, the geometry of the current and reference configurations can not be distinguished. The displacement field of a material point x at time t is u(x, t). The linearized strain tensor is expressed as ε = 1 2 ∇u + ∇ T u and the linearized stress tensor as σ = A : ε, in which A denotes the fourth-order elasticity tensor associated with the constitutive Hooke's law, allowing for the formulation of a differentiable stored strain energy. For any material point x belonging to contact boundary Γ C , we uniquely define, for each x ∈ Γ C , a gap function

Constitutive equations of the continuum

u (x) Ω Γ D Γ C n
g(u) = u(x, t) • n -g 0 (x) (1)
affine with respect to the displacement field u, in which g 0 (x) is the initial positive gap. The contact traction vector τ is decomposed into normal τ N and tangential τ T components according to the outward normal vector n (independent of the displacement field in an infinitesimal deformations framework) as

τ = τ N n + τ T (2)
Accordingly, the boundary value problem describing the motion of the continuum structure parameterized in space and time, may be written as

ρü -div σ(u) = 0 on Ω × R + * (3a) u = 0 on Γ D × R + * (3b) g (u) ≤ 0, τ N ≥ 0, τ T = 0, g (u) • τ N = 0 on Γ C × R + * (3c)
Eq. (3a) describes the local dynamic equilibrium, Eq. (3b) is the Dirichlet boundary condition and Eq. (3c) defines the Signorini boundary conditions, that is: impenetrability, compression and complementary conditions [17]. Any possible inter penetration between the mechanical bodies in presence is precluded by these constraints. We adopt the convention that the "gap" is negative for admissible (i.e. non-penetrated) deformations. Also note that the usual initial conditions are omitted here since they are not relevant for the particular class of periodic solutions of interest.

In the sequel, the derivation of the nonlinear eigenvalue problem is introduced. As eigensolutions are sought in the form of Fourier series, problem (3) will be transformed into a so-called mixed frequency/time eigenvalue problem.

Mixed frequency/time eigenvalue problem

Since we are interested in non-trivial periodic solutions to problem (3), it seems natural to expand the displacement field in the frequency domain using Fourier series:

u(t) = n∈Z ûn e inωt with ûn = 1 T T u(t)e -inωt dt (4)
in which ω = 2π/T is the fundamental nonlinear eigenfrequency of the motion to be determined.

The eigenvalue problem consists in finding {ω, û}, with û = {û n , n ∈ Z} such as,

-div σ(û n ) = (nω) 2 ρû n on Ω × Z (5a) ûn = 0 on Γ D × Z (5b) g (û) ≤ 0, τ N ≥ 0, τ T = 0, g (û) • τ N = 0 on Γ C × [0, T ] (5c)
in which the gap function is now expressed using Eq. (4) as:

g (û) = n∈Z ûn • n e inωt -g 0 (6)
Note that while Eqs. (5a) and (5b) are formulated in the frequency domain, Eq. (5c) requires the use of time, by definition. Accordingly, ( 5) is called a mixed frequency/time domains boundary value problem.

Even though the introduced strong formulation is not of much help for the characterization of a possible solution, it brings light to the intrinsic complexity of the inherent hybrid time-frequency properties of the system due to the direct relationships between the contact tractions subject to spatial geometric constraints themselves dependent on the unknown deformation mappings.

Variational formulation

Strong formulations such as the one detailed above suffer too restrictive assumptions on the class of functions which the solution is sought into. Most of the classical and intuitive conditions of smoothness are not suitable for proving the existence and uniqueness of a solution. Instead, an integral counterpart of system (5) can be derived within the proper mathematical framework of variational formulations.

Constrained minimization of Rayleigh quotient

In this section, the variational formulation associated with the nonlinear eigenproblem defined by Eqs. ( 5) is derived using the concept of Rayleigh quotient minimization. As a preamble, let us define the following set of admissible displacements

V g = {u ∈ V, g (u) ≤ 0 on Γ C } (7)
which accounts for Signorini boundary conditions and is a closed convex subset of the Sobolev space

V = {u : Ω → R d such as u ∈ (H 1 (Ω)) d and u| ΓD = 0} (8)
equipped with the classical scalar product •, • and norm • . Let us also define the usual bilinear and symmetric kinetic and potential energy functionals on V×V to characterize the modal displacements of the system:

m (u, v) = Ω ρuv dx (9a) k (u, v) = Ω σ (u) : ǫ (v) dx (9b)
The following generalized Rayleigh quotient can then be built:

r (û) = k (û, û) m (û, û) (10)
in which the two frequency-domain energy functionals follow from Eqs. [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF] as

k (û, û) = n∈Z k (û n , ûn ) (11a) m (û, û) = n∈Z n 2 m (û n , ûn ) (11b)
Nonlinear eigenvectors û are defined as critical points of this Rayleigh quotient and the associated eigenvalues are such that ω 2 = r (û). Due to Signorini conditions, which confine the set of admissible displacements to (7), this minimization problem is constrained. Furthermore, eigensolutions should be normalized. This is even more important in a nonlinear framework due to the energy dependency of the modal parameters. A possible approach is to normalize eigenvectors with respect to the kinetic energy (this is similar to the usual mass normalization, see [START_REF] Bellizzi | A new formulation for the existence and calculation of nonlinear normal modes[END_REF] for an example with nonlinear modes). We hence define the following functional

h(û) = 1 2 m (û, û) -γ (12)
which relates the kinetic energy to the modal coordinate γ.

Accordingly, the eigenproblem of interest becomes

min ûn∈Vg r (û) subject to h(û) = 0 (13)

Lagrangian formulation

Consider now the following Lagrangian

L (û, λ, µ) = r (û) + T λ(t), g(û) dt + µh(û) (14)
in which λ(t) is a time-dependent and positive Lagrange multiplier field and µ is a (constant) Lagrange multiplier. The initial constrained minimization problem (13) can be converted into a saddle point problem:

min ûn∈V max λ>0,µ L (û, λ, µ) (15)
This Lagrangian formulation enables the use of unrestricted displacement spaces V by transferring the constraints to the Lagrange multiplier dual variables. As a consequence, optimality conditions associated with problem [START_REF] Lesaffre | Stability analysis of rotating beams rubbing on an elastic circular structure[END_REF] are

∇ ûn r(û) + T λ(t), ∇ ûn g(û) dt + µ∇ ûn h(û) = 0 ∀n ∈ Z (16a) g(û) ≤ 0, λ(t) > 0, λ(t), g(û) = 0 ∀t ∈ [0, T ] (16b) h(û) = 0 (16c)
They can be expanded, yielding the following formulation:

Find {û n ∈ V, n ∈ Z} such as ∀v ∈ V 2 m (û n , ûn ) -1 k (û n , v) -n 2 r (û) m (û n , v) + T λ, v e inωt dt + 2µ m (û n , v) = 0 ∀n ∈ Z (17a) g(û) ≤ 0, λ > 0, λ, g (û) = 0 ∀t ∈ [0, T ] (17b) h(û) = 0 (17c)
Similarly to linear systems, solutions of system ( 17) are critical points of constrained Rayleigh quotient [START_REF] Leib | Experimental and Numerical Analysis of Two Vibro-Impacting Beams[END_REF].

Under general yet conservative assumptions, one would have to account for the contribution of nonlinear internal forces in the potential energy of the system and then consider the minimization of the subsequently modified Rayleigh quotient. In the present study, since nonlinear forces arise from contact boundary conditions, their work is identically zero; hence the use of an unmodified Rayleigh quotient to define the respective eigenvalues.

Augmented Lagrangian

The saddle point formulation [START_REF] Lesaffre | Stability analysis of rotating beams rubbing on an elastic circular structure[END_REF] and its subsequent derivations, equivalent to the original nonlinear eigenvalue problem [START_REF] Leib | Experimental and Numerical Analysis of Two Vibro-Impacting Beams[END_REF], are formulated in an augmented Lagrangian framework. As a matter of fact, this formulation improves convergence and computationally benefits from the addition of penalty terms on constraints, allowing for the implementation of efficient Uzawa-like solution algorithms that perform descent on the primal displacement variable, through traditional nonlinear solvers, and ascent on the dual Lagrange multipliers (which coincide with contact forces when convergence is reached). Among all the alternatives, the augmented Lagrangian is here defined as [START_REF] Powell | Algorithms for nonlinear constraints that use lagrangian functions[END_REF]:

L κ (û, θ) = r (û) + 1 2 T √ κ i (g (û) + θ i ) 2 + dt + 1 2 κ e (h(û) + θ e ) 2 (18)
in which κ i θ i and κ e θ e represent Lagrange multipliers for inequality and equality constraints respectively. The diagonal matrix κ i and scalar κ e are respective positive penalty parameters. By first defining the dual parameters θ = {θ i , θ e } and κ = {κ i , κ e }, the principle of such strategies is to generate a sequence of primal iterates û(j) , which minimize the augmented Lagrangian functional [START_REF] Bellizzi | A new formulation for the existence and calculation of nonlinear normal modes[END_REF] for constant multipliers θ (j) , κ (j) which are subsequently updated, in agreement with the constraints. The optimality condition for the minimization of the augmented Lagrangian functional (18) is:

∇ ûn L κ (û, θ) = 0 ∀n ∈ Z (19)
which, according to Eqs. (17), may be written as:

Find {û n ∈ V, n ∈ Z} such as ∀v ∈ V 2 m (û n , ûn ) -1 k (û n , v) -n 2 ω 2 (û) m (û n , v) + T κ i (g (û) + θ i ) + , v e inωt dt+κ e m (û n , v) (h (û) + θ e ) = 0
(20) where parameter θ is then adjusted as follows:

θ i ← θ i + max x,t (g (û) , -θ i ) (21a
)

θ e ← θ e + h(û) (21b)
The penalty coefficients κ may also be updated when conditions [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF] did not improve the constraints.

Numerical implementation and algorithms

In the framework of the finite-element method, we consider a discretized displacement field U together with consistent mass M and stiffness K structural matrices based on [START_REF] Carpenter | Lagrange constraints for transient finite element surface contact[END_REF]. Furthermore, a uniform time discretization is introduced:

t = {t k = kT /m, k = 1, . . ., m} (22)
such as Ūk = U (t k ) are discrete time values of U and the Fourier series of Eq. ( 4) is truncated up to order N . Hence, frequency-domain and time-domain variables are now related to each other in a discrete form:

Ûn = 1 T m k=1 Ūk e -i 2πkn m and Ūk = N n=-N Ûn e i 2πkn m (23)
In practical applications, it is usually required to have N significantly smaller than m. Within these numerical assumptions, the mixed frequency/time augmented Lagrangian eigenvalue problem consists of:

• the discretized Rayleigh quotient used to calculate eigenvalues

r( Û) = N n=-N Û * n K Ûn N n=-N n 2 Û * n M Ûn (24a)
• the gap function (6) as a linear system of equations for each time-step k:

g( Ūk ) = A Ūk -g 0 , 1 ≤ k ≤ m (24b)
where rectangular matrix A restricts the displacement vector U to contact degrees-of-freedom;

• the mode normalization [START_REF] Woo | Application of the harmonic balance method to ground moling machines operating in periodic regimes[END_REF]:

h( Û) = 1 2 N n=-N Û * n M Ûn -γ (24c)
• the eigenvalue equation ( 20) which reduces to 2N + 1 coupled sub-problems:

2M -1 K -(nω) 2 M Ûn + t A m k=1 κ i g( Ūk ) + θ i,k + e i 2πkn m + κ e h( Û) + θ e M Ûn = 0 (24d)
Algorithm 1 Nonlinear modal analysis: simultaneous eigenfrequency/eigenvector resolution 1: Set j = 0, {ω (0,p) , Û (0,p) } 2: for γ (j) in [γ min . . . γ max ] do 3:

Solve the augmented Lagrangian problem (Eqs. (24d) and (24b)-(24c)) and the Rayleigh quotient Eq. (24a) for {ω (j) , Û (j) using Algorithm 3

4:

Predict {ω (j+1,p) , Û (j+1,p) } 5: end for Algorithm 2 Nonlinear modal analysis: sequential eigenfrequency/eigenvector resolution 1: Set j = 0, {ω (0,p) , Û (0,p) } 2: for γ (j) in [γ min . . . γ max ] do 3:

Solve the augmented Lagrangian problem (Eqs. (24d) and (24b)-(24c)) for Û (j) with ω = ω (j,p) using Algorithm 3

4:

Retrieve eigenvalue ω (j) ( Û (j) ) using the Rayleigh quotient, Eq. (24a)

5:

Predict {ω (j+1,p) , Û (j+1,p) } 6: end for Various techniques can be employed to solve the eigenvalue problem defined by Eqs. (24). In particular, we have considered a combined and a sequential approach in which the eigenvalue and eigenvector are found either simultaneously or sequentially as described by Algorithms 1 and 2 respectively. The simultaneous approach is more rigorous yet computationally more expensive (in particular concerning the calculations and updates of Hessians). The sequential approach requires the energy-steps (δh (i) ) to be sufficiently small to ensure that an approximate eigenvalue does not significantly affect the accuracy of the resulting eigenvector. Provided the latter condition is satisfied, the simultaneous and sequential approaches would yield similar results. In both approaches, Eq. ( 24d) is numerically solved using a dedicated augmented Lagrangian algorithm, inspired by the one detailed in [START_REF] Arora | Multiplier methods for engineering optimization[END_REF]. The proposed algorithm 3 offers the following features:

• Convergence is controlled using Powell's measures of convergence:

G i = max dof |max k g Ūk , -θ i,k | (25a) G e = |h( Û)| (25b)
• Updates of multipliers and penalty parameters are determined depending on whether constraints have sufficiently improved, that is to say, for each constraint which does not satisfy:

max g( Ūk ), θ i,k < G i α (26a) h( Û) < G e α ( 26b
)
The two parameters α and β involved in the augmented Lagrangian algorithm should be chosen with care. Although the only theoretical condition is to choose them larger than 1, numerical experience shows that specific values dependent on the problem of interest are necessary to achieve better performances. In the latter applications, α = 4 and β = 10 demonstrated satisfying stability and performances.

The numerical implementation is performed in Python programming language with the help of NumPy/ SciPy [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF] scientific libraries.

Algorithm 3 Augmented Lagrangian with equality and inequality constraints

1: Set j = 0, j max , G (0)
i,e = ∞, θ = θ (1) , κ = κ (1) 2: Choose α > 1, β > 1, ǫ > 0 3: while j < j max do 4:

Solve Eq. (24d) for { Ûn } n=-N,...,N with fixed or variable ω (according to Algorithms 1 or 2)

5:

Evaluate constraints {g( Ūk )} k=1,...,m and h( Û) using Eqs. (24b) and (24c) if Ḡi ≥ ǫ then 12:

Set S i = {l : |max g( Ūk ), -θ i,k | ≥ G (j) i /α} 13: if Ḡi ≥ G (j) i then 14:
∀l ∈ S i , κ i,l ←βκ i,l and θ i,l ←θ i,l /β 15: if Ḡe ≥ G

else 16: θ (j) i ←θ i , κ (j) i ← κ i and G (j) i ← Ḡi 17: θ i ← θ (j) i + max k g( Ū), -θ i 18: if G (j) i ≥ G (j-1) i /α then 19: ∀l ∈ L i , κ i,l ← βκ i,l
θ e ← θ (j) e + h γ ( Û) 29: if G (j) e ≥ G (j-1) e /α then 30: κ e ← βκ e if |h( Û)| ≥ G e /

Applications

This section concerns two applications which illustrate the proposed methodology and algorithms. The first is dedicated to the study a thin rod in contact with a rigid obstacle at one of its boundary and the second focuses on a turbomachinery compressor blade in contact with a rigid contact interface at its tip edge.

Thin rod in contact against a rigid foundation

We first investigate the first nonlinear mode of a simple rod undergoing hybrid force-displacement unilateral constraints, as depicted in Fig. 2. The natural frequency of its linear counterpart with clamped-free boundary The equations of motion can be summarized as:

• local equation: ρ ∂ 2 u ∂t 2 (x, t) -E ∂ 2 u ∂x 2 (x, t) = 0, x ∈ ]0, L [ (27a)
• boundary condition: u(0, t) = 0 (27b)

• unilateral contact conditions:

u(L, t) -g 0 ≤ 0, σ xx (L, t) = EA ∂u ∂x (L, t) ≥ 0, (u(L, t) -g 0 ) • σ xx (L, t) = 0 (27c)
The numerical nonlinear modal analysis is conducted with a separate solution u(x, t) = t N(x)h(t) derived through the usual finite element discretization involving n = 20 rod elements and the following respective elementary mass and stiffness matrices:

M e = ρAℓ e 6
2 1 1 2

and

K e = EA ℓ e 1 -1 -1 1 , ℓ e = L n (28)
After space-integration over the length of the rod, the unknown modal displacement field is given by Eq. ( 4) and the eigenvalue problem derived from Eqs. ( 27) is solved with Algorithm 2. As mentioned earlier, mode normalization of Eq. (24c) through a prescribed kinetic energy controls the amplitude of the nonlinear mode. Fig. 3 displays the first nonlinear mode eigenfrequency with respect to the modal energy (normalized with reference with the initial gap). Contact occurs as soon as the modal energy reaches unity. Then, the natural frequency increases which denotes a hardening effect. Figs. 4 display a portion of the nonlinear internal distribution of the rod nodal displacements over a motion period, namely the displacements of the first and last nodes versus the displacement and velocity of the middle node of the rod. Again, differences between linear and nonlinear regimes are noticeable. While the former typically features elliptic trajectories in the phase space, the latter is much more complex with velocity singularities and quick changes in displacements. Further details about this complex motion can be extracted from Figs. 5, which depict time-histories of all degrees-of-freedom for two normalized modal energies, respectively 0.7 (without contact) and 1.8 (with contact). When the prescribed kinetic energy is low, the usual first mode of the rod is retrieved as pictured in Fig. 5(a)-(b). In this configuration, the mode is a standing wave for which displacement and velocity fields are algebraically related through a phase-shift and an amplitude ratio, as expected. Maximum displacement and (respectively) velocity nodal amplitudes are simultaneously reached. On the contrary, when the prescribed kinetic energy is such that the tip of the rod hits the rigid foundation (normalized energy = 1.8), contact occurs over a fraction of the period of the respective nonlinear mode which becomes a traveling wave, as displayed in Fig. 5(c)-(d). Displacement and velocity fields are no more algebraically related to each other. As a matter of fact, under proper assumptions, Eq. (27a) generically admits a combination of forward and backward travelling wave solutions: these are highlighted by the velocity map histories in Fig. 5d for which the condition of simultaneous maximum nodal amplitudes does not hold anymore. Furthermore, the expected symmetry in time and space of generic travelling waves is also retrieved by the algorithm as pictured in Fig. 5c while no assumptions were made on the displacement field.

A turbomachinery blade in contact at its tip edge

The second application concerns a turbomachinery compressor blade in contact with a rigid contact interface at its tip edge. This situation is representative of rotating compressor components experiencing unilateral contacts with surrounding rigid casings. The finite element model of this blade is shown in Fig. 6 where the red nodes define the contact interface. This model is reduced by means of a Craig-Bampton procedure composed of 72 constraints modes associated with the physical displacements of 24 retained interface nodes belonging to the contact surface Γ C complemented by 40 component normal modes. Contact occurs in the radial direction only and a uniform initial gap is assumed. All displacements and velocities are normalized with respect to this initial gap and nonlinear frequencies are normalized with respect to the first linear frequency of the blade. Fig. 7 depicts the eigenfrequency of the first mode (nonlinear extension of the first bending mode of the blade) with respect to the modal energy. When the normalized energy is greater that one, intermittent contacts occur at the leading and trailing edges of the contact interface. This results in an increase of the natural frequency of the blade.

Figs. 8 display the mode shapes for three values of the modal energy. Sensitivity of mode shapes to contacts is noticeable. In particular, torsion effects tends to appear as modal energy increases and contact becomes more prominent.

This example illustrates a potential application of nonlinear modal analysis in an engineering perspective. Indeed, modal characteristics provide a straightforward on assessment of the effects of contacts on blades' dynamics (change of frequencies, stress concentrations, etc.) One can then image the development of design tools based on nonlinear modal analysis such as nonlinear Campbell diagrams.

Conclusions

A methodology for the modal analysis of elastic structures with contact interfaces is proposed. Based on the definition of nonlinear modes as non-trivial periodic solutions of the autonomous dynamical system, the eigenvalue problem is defined in a continuum framework and involves Signorini boundary conditions. A generalized Rayleigh quotient functional is introduced, which minimization under constraints is formulated using Lagrange multipliers, and solved using an augmented Lagrangian technique. Numerical approximations in space, time and frequency are then proposed for the sake of numerical treatments.

The approach is first illustrated on the example of thin rob in contact at one of its end. Interesting travelling wave motions for a conservative nonlinear system are obtained. Then, a turbomachinery compressor blade in contact with a rigid casing is considered. The first nonlinear mode is studied using the developed numerical tool and the sensitivity of the modal quantities to the contact constraints is explored. Stability and bifurcations analyses of motions of interest may be provided in future research works. Friction effects as well as wear along the contact interfaces can also be implemented.

Nonlinear modal analysis may have long-term applications in the characterization of time-stepping techniques by measuring their capabilities to remain on the respective nonlinear manifolds.

Figure 1 :

 1 Figure 1: Notations With reference to Fig. 1, we consider a flexible body occupying the open set Ω of R d (d = 2 or 3) which boundary Γ consists of mutually disjoint open manifolds Γ C (contact boundary), Γ D (Dirichlet boundary) and Γ F (Neumann boundary) such that Ω = Ω∪Γ. Since the system of interest is autonomous, the prescribed external forces are identically zero.In the framework of infinitesimal deformations, the geometry of the current and reference configurations can not be distinguished. The displacement field of a material point x at time t is u(x, t). The linearized strain tensor is expressed as ε = 1 2 ∇u + ∇ T u and the linearized stress tensor as σ = A : ε, in which A denotes the fourth-order elasticity tensor associated with the constitutive Hooke's law, allowing for the formulation of a differentiable stored strain energy. For any material point x belonging to contact boundary Γ C , we uniquely define, for each x ∈ Γ C , a gap function

6 :

 6 Set Powell's criteria Ḡi and Ḡe according to Eqs. (25) 7:if Ḡi < ǫ and Ḡe < ǫ then

  e ←βκ e and θ e ← θ e /β if h( Û) < G e /α e ← θ e , κ (j) e ← κ e and G (j) e ← Ḡe 28:

Figure 2 :

 2 Figure 2: Thin rod in contact against a rigid foundation

Figure 3 :

 3 Figure 3: Frequency-energy plot of the first nonlinear mode

Figure 4 :

 4 Figure 4: Periodic motions of the first nonlinear mode represented in a three-dimensional projection of the phase space; Displacements of the (a) last and (a) first nodes versus displacement and velocity of the middle node.

Figure 5 :Figure 6 :

 56 Figure 5: Time-histories of rod's displacement and velocity degrees-of-freedom for normalized energy 0.7 ((a) and (b)) and 1.8 ((c) and (d))

Figure 7 :

 7 Figure 7: Frequency / energy plot of the first nonlinear mode of the blade

  α

	31:	end if
	32:	end if
	33:	end if
	34:	end if
	35: end while

Acknowledgements

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.