Towards an Art of Impregnation

In The Routledge Companion to Sounding Art. Marcel Cobussen, Vincent Meelberg & Barry Truax (Eds.) New York: Routledge, 2017, p. 265-273

Jean-Paul Thibaud

There is increasing common ground between the art world and the social sciences. This is particularly true of work related to sonic spaces, sensory environments and everyday aesthetics. Shared questions are surfacing, enabling us to take advantage of cross-breeding and the contribution of the two perspectives. The field of urban ambiances has featured largely in these exchanges between artistic creation and scientific research. Whether one defines ambiance as the sensory atmosphere of place, the pervasive quality of a situation or the affective tonality of an environment, the question of in situ experience is inevitably raised. This means that an ambiance is always situated, set in a particular space and time, which it helps to singularize and qualify in return. Consider the buzz of a market or railway station, or the way an airport, cathedral or public square sounds. A singular atmosphere emerges from each of these spaces, expressing and inducing a way of being together.

How does this relate to the sonic background which accompanies us in time and in which we take part? In what way do such ambiances affect us in our daily lives? How may we open our ears to the atmospheric existence of the contemporary world? In asking such questions I intend to introduce the notion of ambiance in the field of sonic studies. My aim is to focus on the overall salience of ambient sound and to better understand how it infuses and pervades everyday urban experience.

My purpose is not to define the notion of ambiance but rather to query its efficacy, to cast light on the effects it produces and its hidden potential. So this involves both acknowledging the intangible, evanescent character of ambiance while underlining its power to act and transform. After all, we may say that an ambiance is moving, calming, disturbing or compelling. In short, though ambiance is difficult to grasp and circumscribe, it nevertheless remains pregnant and effective. The main point I wish to develop is that ambiance proceeds by exerting an unobtrusive but lasting influence on anyone experiencing it. Far from being peremptory or injunctive, it operates through influence, as

From immersion to impregnation

We should certainly start by acknowledging the power of the sonic world, as pointed out by Walter J. Ong (1981: 112) To answer these questions I propose to explore impregnation, the discreet, pervasive and durable mode of action by which ambiance takes effect. As a preliminary approximation, impregnation refers to modus operandi by which an ambiance accomplishes its immersive power. Ambient music is a perfect example of a sensory medium's capacity for affecting us almost imperceptibly and leaving its mark on our ways of being in situ. If we accept that any place is qualified by an affective tonality, prevailed over by an ambiance, then impregnation is the formula for such prevalence. So our task is to describe how an atmosphere infuses a sense of place and leaves marks in the inhabitant experience. This is no easy matter because impregnation operates through a fragile relation with the ambiant world, a thing of nuances, gradations and restraint. It is a question of je-ne-sais-quoi and presque-rien [START_REF] Jankélévitch | Le je-ne-sais-quoi et le presque-rien[END_REF], hard to put into words and clearly enunciate. A whole lexicon could be compiled in an effort to express this tenuous, yet vital relation with the ambiant milieu: from floating to touching lightly, exhaling to inclination, incitation to infiltration, allusion to evocation. Much as when a novel carries us away to another world on a stream of words, or when music gently picks us up with its rhythm. For a moment we are living in the novel, we belong to the music. Here it is all a matter of "tiny bonds" [START_REF] Laplantine | De tout petits liens[END_REF]. Clearly, impregnation is not a matter of immediate, manifest efficacy; rather it plays on our porosity to the ambiant environment, our ability to enter into resonance with a sensory space.

But in that case how does sounding art help us to pay attention to a sensory atmosphere's tiny attachments and incitements? Of course, we can also focus on a city's most remarkable and spectacular sonic events, such as street parties and collective entertainment of all sorts. However we are more interested here in the infra-ordinary [START_REF] Perec | L'infra-ordinaire[END_REF], in the evidence of the already-there and the ever-present, in recognizing the efficacy of a background sound, the existence of an acoustic backdrop which accompanies us in our everyday dealings, without us necessarily paying much attention. In fact our concern here is to address the "noise of existences" [START_REF] Sansot | Existence des bruits et bruits des existences[END_REF]). Murray Schafer quite rightly identified this sonic domain made up of the thousand and one small noises of everyday life, too ordinary really to be noticed: the hum of air-conditioning, scraping of doors, pitter-patter of steps and whispering voices, the whish of human gestures and technical devices, all sorts of rubbing and brushing. In proposing the notion of keynote sounds he emphasized their subconscious, but no less pregnant character: "even though keynote sounds may not always be heard consciously, the fact that they are ubiquitously there suggests the possibility of a deep and pervasive influence on our behaviour and moods" (Schafer 1977: 9). In focusing on the sonic impregnation of our living environment, my aim is to take Schafer's initial proposition a bit further and put it to the test by trying to tease out its possible consequences.

Testing impregnation against music

The world of music offers valuable clues to progress with investigation of the imperceptible efficacy of urban ambiances. Several contemporary philosophers have gone out of their way to show how music deploys unsuspected powers. Whether it is a matter of "giving ear to the unheard," as Michel [START_REF] Serres | Musique et bruit de fond[END_REF] Lending an ear to a background and to forces at least as much as to form, these ways of listening to the ambiant world undoubtedly bring us closer to the work of impregnation.

These two conceptsbackground and forceare essential to any attempt to conceptualize the efficacy of ambiance. On the one hand the sonic ambiance is not so much a set of sounds which can easily be discretized, listed and identified, rather an immediately experienced overall tonality (Thibaud 2011a). In short any ambiance proceeds from a qualified sensory background, the content of which we need to gauge, particularly in an urban environment. On the other hand, the sonic ambiance has the capacity to affect us and presents itself as a domain of an intensive order. If the ambiance has an influence it is precisely because it exerts forces on anyone experiencing it, in this case city-dwellers. By highlighting the nature of these backgrounds and forces, the notion of impregnation will really begin to make sense. So what can be said of sonic impregnation in/of an urban environment?

Ambiance as tonalization

We should start by saying that sound enables us to draw attention to the fundamental experience of resonance. Not only does it mark the sonority of a place, but it is the basis for our ability to reverberate with and be affected by the ambiant world [START_REF] Nancy | On Listening[END_REF]). We should therefore accept that we are not outside of, or indifferent to our surroundings, rather that we resonate with, in and through them. Resonance is not a simple acoustic phenomenon but more fundamentally refers to our ability to vibrate in unison with the world [START_REF] Minkowski | Vers une cosmologie[END_REF]. It indicates the entanglement of world and body and emphasizes our power of attunement.

A brief detour via the music of Claude Debussy may help us to clarify this difficult point. His work may be seen as the starting point for an aesthetics of sonority [START_REF] Guigue | Esthétique de la sonorité. L'héritage debussyste dans la musique pour piano du XXème siècle[END_REF], breaking free from formal continuity and melody by creating chords for their timbral effect, with compound resonances, and focusing on pure instants of becoming. He thus enabled us to hear overall sonic states endowed with a particular atmosphere. We are dealing with blocks of sensation, which evoke rather than narrate, suggest rather than recite, impress rather than represent. Jankélévitch puts it beautifully: "Music does not express [an idea] word for word, nor does it signify point by point, but on the whole suggests; it does not lend itself to juxtalinear translations nor to confiding indiscreetly intimate details, but rather to atmospheric, pneumatic evocations" (1983: 69). Debussy uses all his timbral harmony to find sonic colors. But what interests us here is that his work may serve as a key for understanding impregnation.

Gaston Bachelard, who proposed the notion of tonalization, used the metaphor of the Aeolian harp, drawing much of his inspiration from Debussy to show how we enter into sympathy with our ambiant milieu [START_REF] Bachelard | La poétique de l'espace[END_REF]). Debussy's evocation of the sea, in La Mer, is so successful, not because it imitates as closely as possible the real noises of the sea and waves, but because it enables us to hear the motor premises, the climatic forces and the qualities of the flux driving an atmospheric experience of this sort. Marie-Pierre Lassus sums this up very well in her book on Bachelard and music: "Music does not imitate the sound of wind or water but prompts a form of bodily empathy which makes us sketch out in our minds movements which stir reminiscences linked to our singular experience of the landscape" (2010: 55). With Debussy and Bachelard we can understand how sonic moments spark our motor imagination, trigger qualities of motion and initiate "vitality affects" [START_REF] Stern | Forms of Vitality: Exploring Dynamic Experience in Psychology, the Arts, Psychotherapy, and Development[END_REF]). If impregnation is at work here, it is through tonalization, reverberating with the vibrations in the ambiant world.

Pinpointing the tonalization phenomenon enables us to distinguish between environment and ambiance. One way of putting it would be to say that an ambiance is what gives life to an environment, confers its emotional value and appeal. Any lived space is marked by this pathic relation which we entertain with the world around us. In urban terms, ambiance operates by setting the tone of territories, endowing them with both an affective tonality and a particular overall physiognomy. As a result a single adjective is sufficient, enabling us to qualify an urban situation as joyful, gloomy, electric, friendly, peaceful or stressful. In this respect Jean Ladrière sees the city as an "existential inductor, in so far as we can recognize in it, through resonance, a certain affective tonality which qualifies our existential movement" (1973: 155). Clearly the idea of tonalization is heavy with consequence; it is in this way that existential territories settle in. Sound is of prime importance in this process, if one considers that setting the tone is much the same as making a voice heard which ushers in an atmosphere and invokes a quality of situation, expresses an affective disposition and makes a mood audible. There is no question here of imposing the reign of a dominant tonality which would mark the urban world as a whole, but rather of asking how the sensory fashions territory and querying the tonalizing powers of inhabited spaces. Whereas modernist metropolises were marked by hyperstimulation [START_REF] Simmel | The Metropolis and Mental Life[END_REF]) and shock aesthetics [START_REF] Benjamin | The Writer of Modern Life: Essays on Charles Baudelaire[END_REF], contemporary Western mega-cities tend to be described in terms of detachment from reality, leading to an uncanny feeling [START_REF] Vidler | The Architectural Uncanny: Essays in the Modern Unhomely[END_REF], an almost hallucinatory experience [START_REF] Bégout | Zéropolis[END_REF], anaesthetized perceptions [START_REF] Ingersoll | Sprawltown: Looking for the City on its Edges[END_REF] or a preponderance of feeble sensations [START_REF] Koolhaas | [END_REF].

Ambiance as atmospherization

A second way of questioning the powers of ambiance is to experiment with the thresholds of perception. Many musicians and sound artists focus on these tiny perceptions, which are so infinitesimal and mixed up with one another that we do not usually perceive them distinctly. Leibniz liked to take as an example the roar of the sea made up of the sound of an infinite number of barely audible ripples, which we would have difficulty picking out but which nevertheless affect us. "We must be affected slightly by the motion of this wave, and have some perception of each of these noises, however faint they may be; otherwise there would be no perception of a hundred thousand waves, since a hundred thousand nothings cannot make something" (1996 [1704]: 55). He makes clear that not being fully aware of these tiny perceptions does not prevent us from being affected. From this point of view, impregnation is not a question of manifest phenomena but rather a low-intensity influence which infiltrates itself on all sides and takes hold over time. What is difficult is to give meaning and existence to this slight register of sensory experience, which Marcel Duchamp referred to as the "infra-mince" (infra-thin). It is very much as if this order of reality is too minimal to be noticed, too evanescent to grasp, to insignificant to be worth of attention. Yet these tiny fluctuations in perception weave the fabric of our familiarity with the ambiant backdrop, which modulates our state of being and our background feelings [START_REF] Damasio | Spinoza avait raison. Joie et tristesse, le cerveau des émotions[END_REF].

Here again we may be guided by sound. The 20th century witnessed a steady drive towards the "molecularization of musical material" (Critton 2011), working on threshold effects and micro-variations, liminal phenomena and micro-intervals. Concepts such as spectral music, granular synthesis, musical continuum and micro-tonality spring to mind.

All of these heighten our ability to sense tiny variations in sonic matter and explore the fringes of sensory experience. Their aim is to make us hear the way one shifts imperceptibly from one state to another. In this respect Pierre [START_REF] Boulez | Penser la musique aujourd'hui[END_REF] draws a distinction between the smooth space which "occupies without counting," a fluid, moving space populated by forces and directions, and the striated space which "counts in order to occupy," a metric, hierarchical space made up of shapes and dimensions. Using music to reveal this smooth space opens the way for the conception of an intensive space which really gives full scope to affects [START_REF] Deleuze | Mille Plateaux[END_REF]. In this case impregnation proceeds imperceptibly and continuously, below a level which we are capable of representing to ourselves, thanks to non-polarized affects and confused sonic impressions, and in constant motion. It certainly seems appropriate to refer to this as "atmospherization" in so far as the affective investment does not apply to a specific object but to an environment as a whole [START_REF] Tellenbach | Goût et Atmosphère[END_REF].

One important consequence this involves is to emphasize the importance of continuous flows rather than discreet signs in establishing a space's ambiance. This is how it is possible to distinguish between context and atmosphere. As Fernando Gil notes: "The former

[context] is visible and can be reduced to a set of relations and signs. It is semiotizable, whereas the atmosphere is infra-semiotic, unfolding in a continuum" (2000: 16). So an urban ambiance proceeds in a pervasive manner, without it being possible to pick out a single, specific origin for the affects deployed. Should one nevertheless hazard a guess at the factors triggering an affective tonality, it is only possible to draw up a partial, necessarily incomplete list. For example, as Annie Honnorat and Pierre Sansot explain, a happy street is due to "the welcoming attitude of the people you meet, their accents, the cut of their features, surprising colours which never shock the eye, nimble gestures, shop displays, helpful staff, graceful buildings and the clarity of the overall design" (1998: 130). This list could go on forever, not only because allowance must be made for other components (the weather, the pace of movement, etc.), but above all because this process of atmospherization is fundamentally pervasive, continuous and infra-semiotic.

This does not mean that ambiances are all similar and have the same intensive strength.

As Gilles [START_REF] Deleuze | Le Pli. Leibniz et le Baroque[END_REF] emphasized, small perceptions are not about the relationship between the parts and the whole but between the ordinary and the noticeable. That's really what urban ambiances are all about, in other words, their respective degree of salience and consistency. Some sound atmospheres seize us more than others, some are more sensitive to the expressive power of inhabitants (Thibaud 2011b). If any sonic ambiance convenes a set of microperceptions that set their mark on the places and "atmospherize" them, each demonstrate a specific capacity to affect city dwellers and be affected by them. Hence, impregnation is not only a matter of the nature of affective tonalities but also of their force and intensity. In other words, ambiances are distinguishable according to their respective power of impregnation.

Ambiance as climatization

A third approach to questioning the impregnation of an ambiance starts with the observation that music has become omnipresent in daily life, in retail spaces and transport facilities, in leisure centres and workplaces. It can hardly be a coincidence that two authors [START_REF] Toop | Ocean of Sound: Ambient Music, Imaginary Worlds and Aether Talk[END_REF][START_REF] Prendergast | The Ambient Century: From Mahler to Trance[END_REF]) should have coined the term "ambient music" to qualify the 20th century, on account of its enveloping, ubiquitous character. This calls to mind the work of both Muzak and Brian Eno, on account of their ability to create particularly immersive sound environments. There is no shortage of terms referring to these types of music, in which we are constantly immersed and which, most of the time, we blank out from our awareness, terms such as functional music, elevator music, furniture music, background music or easy-listening music. Regardless how diverse their aesthetics and finalities, their common purpose is to broadcast music in selected urban environments, and to compose it for specific times, activities and places. Ambient music starts from the idea that it is possible to use music to instil a particular atmosphere and to modulate the mood of the target audience almost subliminally. It would certainly be misleading to lump together all forms of music under this heading. Eno himself makes this point, distinguishing his work from Muzak's, even if there is common ground between the two.

Whereas Muzak "is meant to be heard, but not listened to," Eno's music tries to lend itself to various types of listening. As he explains in the liner notes on Music for Airports Ambient 1 1978), "Ambient Music must be able to accommodate many levels of listening-attention without enforcing one in particular; it must be as ignorable as it is interesting."

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Over and above these differences ambient music leads us to the question of lack of attention which is central to the sonic experience. This means that the efficacy of a musically instilled ambiance does not necessarily depend on focused or attentive listening.

Indeed it might be quite the contrary: listening in a drifting, peripheral, distracted or unfocused way affects us just as much, and more durably and subconsciously. Muzak makes no bones about this, offering to orchestrate and modulate the public's affects by operating below the level of its awareness [START_REF] Sumrell | The Stimulus Progression: Muzak and the Culture of Horizontality[END_REF]. Our capacity to be affected also operates without us being vaguely aware of it or expressing any intention, as is the case in daily life governed by heedlessness and what [START_REF] Benjamin | L'oeuvre d'art à l'ère de sa reproductibilité technique[END_REF] refers to as perception "through habit."

The increasing musicalisation of the urban environment often goes hand-in-hand with other systems such as air-conditioning and artificial lighting. In every case the aim is to create artificial sensory environments and to climatize living milieus. Here ambiance is a matter of climatization, defined as "the daily operation of artificial, yet invisible conditioning by which humans create their own ambiance for living, forming relatively closed, protected spaces encased in tradition and social life, where they can breathe the healthy, innocent air of familiarity" (Bégout 2005: 260). The sonic ecology of urban spaces is thus increasingly the focus of intervention and design. Peter [START_REF] Sloterdijk | Ecumes. Sphères[END_REF] has probably taken analysis of this process of climatizing our living spaces further than anyone else. In his Spheres trilogy he shows that human existence only really makes sense when seen in the context of the atmospheres which make it possible. We live our lives in spheres, moving from one sensory envelope to another, developing climatic installations to suit our needs. Until now this climatic condition for our existence was largely implicit, but it is now becoming explicit. As Dominic Desroches (2010) explains, summing up Sloterdijk's position, humans are becoming "designers of atmospheres." Sound plays an important part in this process, as the means by which a group is brought into step, bonds internally and is impregnated by ambiance: "The first human island is covered by a psycho-acoustic bell-jar, like some shopping mall with Christmas music. It constitutes its sonospheric cohesion through the undulating presence of voices and noises with which the group impregnates itself as a proprioceptive entity." (Sloterdijk 2005: 334).

Two further aspects of the efficiency of an ambiance are revealed from climatization.

On the one hand, ambiance is more effective and assertive when it involves long periods of exposure that eventually form into habits. A lived everyday sound environment generates percepts and forms ways of hearing that fully participate in a social form of life. In other words, impregnation is a phenomenon that persists and endures over time by gradually becoming naturalized. On the other hand, climatization can be interpreted in terms of "encapsulation" (de Cauter 2004) insofar as the creation of artificial environments tends to produce standardization, privatization and asceptization of urban spaces. This more critical version of climatization reminds us that sonic design is eminently political and engages urban issues of the utmost importance: tendency to evacuate urbanness and to neutralize the public character of urban space, impoverishment and normalization of sensory experience, manipulation of attention for commercial purposes, etc. It is the whole issue of ambient sound as a common good which is at stake here.

Conclusion

Approaching urban ambiances through sound opens a new perspective about in situ experience. The question is not only to recognize the immersive power of sound but also to emphasize its ability to permeate experience and to affect lives. The sonic experience of an ambiance does not uniquely involve focused and conscious perception but also global and diffuse sensations more difficult to grasp. An art of impregnation is at stake, which relies on the resources of sound and music to understand the atmospheric existence of the urban world. In this regard, three joint movements have been described: tonalization qualifies urban areas in terms of affective tonalities and activates the ability of the body to resonate to the vibrations of the surrounding world; atmospherization transforms urban spaces into intensive milieus and pervades experience with small perceptions; climatization converts urban spaces into conditioned environments and frames the social forms of sensitivity. This perspective makes it possible to introduce background and low intensity phenomena into sensory experience of urban life. An art of impregnation makes sense through sound, making urban atmospheres a fragile and nuanced area of experience, and attentive to the discrete efficiency of ambient phenomena.

  : "Sound signals the present use of power, since sound must be in active production to exist at all." From the most remote mythology to contemporary experiments there are numerous accounts of the power of sound. Think for instance of the spell worked by the song of the Sirens in the Odyssey, of noise which subjugates or music which enchants, of the sound environments designed to encourage consumption or to distract us, of the use of sound in medical treatment or warfare, of the capacity of sound to mark out a territory or bring a marching crowd into step. Sound displays a capacity to take possession of bodies and to acclimatize inhabited spaces. But although it is now accepted that sound plays a part in an immersive experience, we are still a long way from describing or explaining the processes at work in such sonic immersion.Work on this topic generally emphasizes the fact that sounds come from all sides, enveloping us, even passing through and penetrating us. But in what way do they transform us? In what manner are we inhabited by the sonic environments we experience?

  puts it, or "rendering audible forces [which are] inaudible in themselves" as Gilles[START_REF] Deleuze | Rendre audibles des forces non-audibles par elles-mêmes[END_REF] stated, it does seem that music has the power to give us access to almost imperceptible phenomena. Serres cites the music of Iannis Xenakisin particular Pithoprakta (1956)to show how it succeeds in making the acoustic background of the universe audible. An operation of substitution is at work here by which "the continuum of the course replaces the momentary discontinuum of the note" and "background without form replaces form without background."(Serres op. cit.: 187). What the music enables us to hear are the conditions under which the world is audible. For his part, Deleuze emphasizes the emergence of a beatless tempo, released from the constraints of measure. Although this idea slightly differs from what Serres suggests, substitution is nevertheless at work here too, giving preference to "sonic molecules rather than pure notes or tones." Here again the question of form becomes secondary, because the "pair constituted by material and forces supplants matter and form."(Deleuze op. cit.: 145). So this time imperceptible forces are discovered and brought to the fore. In both cases music makes audible what cannot be heard without it.


B Hervé

J.-J Sinou

H Mahé

L Jézéquel

Extension of the Destabilization Paradox to Limit Cycle Amplitudes for a Nonlinear Self-Excited System subject to Gyroscopic and Circulatory actions

Keywords: friction, destabilization paradox, limit cycles, self-generated vibrations

This study aims at clarifying the phenomenological roots of an acoustical disturbance known as "clutch squeal noise". A nonlinear two-degrees-of-freedom model is introduced in order to illustrate some basic phenomena leading to self-generated vibrations. The damping of the system as well as both circulatory and gyroscopic actions are included in order to highlight their respective influence and the destabilization paradox. Results are obtained on the stability range of the equilibrium, the nature of the Hopf bifurcation, the limit cycle branches and their stability. A dynamic extension of the destabilization paradox is proposed and some non-periodic behaviours are identified too.

Introduction

In cars with manual transmissions, different unforced vibrations can be observed during the sliding phase of clutch engagement. Low frequencies phenomena such as judder can often be attributed to misalignment in the transmission chain or to tribological properties of the friction materials, such as a decreasing friction coefficient with regard to the sliding speed. However, noise due to high frequencies phenomena (up to few kHz) can be experienced too and is referred as "clutch squeal noise". This audible disturbance can arise even when the friction coefficient is almost constant and the transmission chain shows no important fault. The observed amplitude of the phenomenon allows excluding the assumption of a stick-slip cycle too. The present study aims at investigating an instability origin of these vibrations related to the non-conservative action of the friction forces. The non-conservative aspect of the friction forces is known to be a potential cause for the rise of such selfgenerated vibrations in dynamical systems through the destabilization of their stationary states. This has motivated numerous researchers to focus on such mechanisms for years, see [START_REF] Kinkaid | Automotive disc brake squeal[END_REF]. Some other industrial applications in need of such studies can be cited without being exhaustive: brakes, machining, civil engineering and aeronautic design. In general, a stability analysis only requires the knowledge of the linearized equations of motion in the vicinity of the equilibriums, see [START_REF] Sinou | Friction induced vibration for an aircraft brake system[END_REF]. Therefore, phenomenological roots of the spontaneous rise of motion can be highlighted through rather simple considerations. However, the price to pay for this simplicity is the complete ignorance of the nature and of any property of the steady state which then takes place. This lack of knowledge can lead to misinterpretation of experimental observations as well as bad or excessively stringent design recommendations. The present paper aims at demonstrating the important role of the nonlinearities and the damping structure in dynamical systems subject to flutter destabilization. To this purpose, a paradigmatic example model of non-conservative coupling between the degrees of freedom of a clutch is proposed in the first part. This nonlinear model shows two degrees of freedom coupled by both circulatory and gyroscopic actions and takes into account the structural damping of the system. The second part serves as a reminder about fundamental results on the stability of the stationary states. Considerations for the damping structure of the system lead to counter-intuitive results and the so-called "destabilization paradox". This phenomenon was first reported by Ziegler in [START_REF] Ziegler | Die Stabilitätskriterien der Elastomechanik[END_REF]: he demonstrated the significance of dynamic terms and the destabilizing effect of weak damping for a double pendulum subjected to tangential load. Further developments were proposed by Bolotin [START_REF] Bolotin | Nonconservative problems of the theory of elastic stability[END_REF] for non-conservative stability problems. Then, some researchers (Hermann et al. [START_REF] Herrmann | On nonconservative stability problems of elastic systems with slight damping[END_REF], Leipholz [START_REF] Leipholz | Stability of elastic systems[END_REF] and Kounadis [START_REF] Kounadis | On the paradox of the destabilizing effect of damping in non-conservative systems[END_REF]) studied the destabilizing effect of viscous damping for mechanical systems subjected to non-conservative forces. Moreover, Thomsen [START_REF] Thomsen | Chaotic dynamics of the partially follower-loaded elastic double pendulum[END_REF] examined the non-linear dynamics of a double pendulum with both linear damping and non-conservative follower loading. He studied the occurrence of chaotic motion and changes in amplitude due to a destabilizing effect of both linear and non-linear forces. The influence of the velocity-dependent forces on the stability of non-conservative systems as well as the effects of non-linearities were also studied by O'Reilly et al. [START_REF] O'reilly | Some aspects of destabilization in reversible dynamical systems with application to follower forces[END_REF] and Kirillov and Seyranian [START_REF] Kirillov | Stabilization and destabilization of a circulatory system by small velocity-dependent forces[END_REF]. In spite of these intensive investigations, this topic still motivates researches aiming at understanding the complex destabilization mechanisms in presence of damping, see for example [START_REF] Kirillov | Destabilization paradox due to breaking the Hamiltonian and reversible symmetry[END_REF][START_REF] Hoffmann | Effects of damping on mode-coupling instability in friction induced oscillations[END_REF][START_REF] Sinou | Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping[END_REF][START_REF] Fritz | Investigation of the relationship between damping and mode-coupling patterns in case of brake squeal[END_REF][START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF]. A static influence of the nonlinearities is also evocated in the determination of the equilibriums and the nature of the Hopf bifurcation points by the analysis of the first Lyapounov coefficient. The important role of the structural damping is clearly illustrated and a particular relationship to the gyroscopic action is highlighted. The third part introduces a nonlinear method for determining the limit cycles of autonomous mechanical systems which is appropriate to perform parametric studies. An application of the Floquet theory is also described in order to determine the stability nature of the identified periodic solutions. Then, this approach is applied on the clutch model to investigate its limit cycles. The complex dynamic behaviour of nonlinear structures is clearly illustrated on this example. Neimark-Sacker bifurcations and a possible route to chaos according to the Ruelle-Takens scenario are highlighted with the help of Poincaré sections. Such a turbulent behaviour was already mentioned in [START_REF] Thomsen | Chaotic dynamics of the partially follower-loaded elastic double pendulum[END_REF] for instance. Finally, conclusions are drawn on the sensitivity of self-generated vibrations in mechanical structures to the physical parameters. A particular attention is paid to the role of the structural damping and the gyroscopic actions both on the stability of the equilibriums and the post-bifurcation dynamic behaviour.

2-DOF nonlinear model of a squealing clutch

The proposed model of clutch is depicted on Fig. 1. This system was first proposed by Wickramarachi in [START_REF] Wickramarachi | Analysis of friction-induced vibration leading to EEK noise in a dry friction clutch[END_REF] and extensively investigated in the proposed simplified form depicted on Fig. 1 by the authors in [START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF]. It consists of two discs (A) and (B) coaxially rotating about the Oz axis and rubbing on each other. Oxyz is the principal inertia frame of (A). (A) is the friction disc and (B) represents the engine flywheel and the clutch. Two degrees of freedom (DOF) are considered. They correspond to the swinging motions of (A). The swinging angles about Ox and Oy are noted θ and φ respectively. Restoring forces about these motions are represented by linear stiffness and damping elements, (k θ , d θ ) and (k φ , d φ ) about Ox and Oy respectively. Friction is modeled in a simplified manner: four deformable elements are considered, equally distributed on radius R on (A), and friction occurs at their end. They represent the flexibility of the contact area and they are assumed to show equal linear stiffness and damping (k c , d c ) about Ox. A constant Coulomb law is considered with the constant friction coefficient µ. The rotation speed of both the discs is assumed to be constant. The parallelism of the discs (θ = 0 and φ = 0) illustrated on Fig. 1 supposed to correspond to equilibrium. The distance between the Oxy plane and the end of the contact elements is noted h. Finally, the amplitude of the vibrations is assumed to be small enough so that the friction force at the end of each contact element never vanishes. The following nonlinear nondimensional equation of motion can be obtained for this system,

is R x y z h O (A) (B) φ ϖ Ω θ φ φ d k , θ θ d k , 4x k c , d c
θ φ + 2ξ ρ -ρ 2αβξ θ φ + 1 ϕ -ϕ α 2 θ φ + σ θ 3 φ 3 = 0 0 . ( 1
)
The upper dot denotes time differentiation with regard to a nondimensional time scale τ . ξ ≥ 0 is the reference damping factor. α ≥ 1 and β ≥ 0 are the ratios of the natural frequencies and damping factors respectively. ϕ and ρ are the nondimensional circulatory and gyroscopic factors respectively. Finally, σ > 0 is related to the local stiffening property of the system in the case of nonvanishing motions. These nondimensional parameters are related to the physical parameters with the expression reported in Eq. 2

                                                 ω 0 = 1 J 2R 2 k c + k θ - N 0 2 h (2 + µ 2 ) + (J ̟ -J ) ̟ 2 ξ = 1 2ω 0 1 J 2R 2 d c + d θ + µ N 0 2 h 2 R |̟ -Ω| ρ = 1 ω 0 1 J (2µRhd c sign (̟ -Ω) + (J ̟ -2J ) ̟) ϕ = - 1 ω 2 0 1 J µR N 0 2 1 + h 2 R 2 -2hk c sign (̟ -Ω) α = 1 ω 0 1 J 2R 2 k c + k φ - N 0 2 h (2 + µ 2 ) + (J ̟ -J) ̟ 2 β = 1 ξ 1 2αω 0 1 J 2R 2 d c + d φ + µ N 0 2 h 2 R |̟ -Ω| , ( 2
)
where t is the natural time scale, J is the swinging inertia about Ox and Oy, J ̟ is the rotational inertia about Oz, ̟ is the rotation speed of (A), Ω is the rotation speed of (B) and N 0 is the load applied to the system.

A particular symmetry of Eq. 1 can be observed since if θ φ T is a solution for a given value of (ϕ, ρ) = (ϕ 0 , ρ 0 ) then θ -φ T is a solution for (ϕ, ρ) = (-ϕ 0 , -ρ 0 ). Therefore, the investigation of negative values of ϕ can be omitted in any parametric investigation. One can also note that the amplitude of σ has no qualitative effect on the solutions since if θ φ T is a solution for σ = σ 0 then σ 0 /σ θ φ T is a solution for any other value of σ. 0 0 T is the unique equilibrium of Eq. 1. Finally, the linearization of Eq. 1 in the vicinity of the equilibrium simply gives

θ φ + 2ξ ρ -ρ 2αβξ θ φ + 1 ϕ -ϕ α 2 θ φ = 0 0 . (3)
The values used for numerical illustrations in this paper are α = 1.5 and σ = 10 4 . As highlighted in [START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF] the value of α has no qualitative effect on the stability analysis, provided α = 1. Therefore, the choice α = 1.5 is representative of any assumed difference between the static frequencies of the swinging modes for the stability analysis. Moreover, for a fixed value of α, the value of σ has no qualitative effect neither, as mentioned later. The choice of σ = 10 4 allows obtaining reasonable amplitudes for the limit cycles. The influence of the values of both α and σ together on the nonlinear behaviour is not analyzed in this paper. Parametric investigations are performed on the other parameters.

Investigation of equilibriums

The equilibriums are steady static states of a system. Their investigation aims at understanding the reasons why a dynamic state arises preferentially to equilibrium. This section is divided into three parts. Firstly, the eigenvalues approach of the stability analysis is introduced in order to highlight how equilibrium can become a repulsive state. Secondly, a local insight into the nonlinear properties of the system in the vicinity of bifurcations is proposed in order to describe the loss of stability and explain jump phenomena, i.e. the sudden rise of a large motion when crossing the stability frontier. Finally, the stability of the equilibrium of the proposed model is investigated and important information is obtained on the role of damping and gyroscopic actions.

Stability analysis

Consider the equation of motion of a dynamic system

Ẋ = f (X, p) , (4)
where X represents the instantaneous state of the system (its coordinates in the phase space), the upper dot denotes time derivation and f is a function of X parameterized by the elements of p. Assume at least one stationary solution X e (p 0 ) exists for p = p 0 , i.e. f (X e (p 0 ) , p 0 ) = 0, and f is C 1 in its vicinity. Then, according to the Hartman-Grobman theorem, the linearization of Eq. 4 in the vicinity of X e (p 0 ) preserves its non-marginal stability nature. Therefore, the determination of the stability nature of equilibriums only requires the knowledge of the linearized equation of motion in their vicinity in most cases.

Nevertheless, the nonlinearities allow many equilibriums to exist simultaneously and their evolution when varying p can be rather complex and possibly discontinuous. Therefore, all equilibrium branches have to be determined by appropriate approaches as a first step prior to perform a stability analysis of each branch separately. Assume the linearized form of Eq. 4 expressed in Eq. 5,

Ẋ ≈ D f (X e (p 0 ) , p 0 ) X. (5)
The solutions of such a linear system form a vector space whose basis is constituted of elements of the form expressed in Eq. 6 when the Jacobian is diagonalisable, where X 0 (p 0 ) is the mode shape, t the time and λ (p 0 ) a complex factor. λ (p 0 ) and X 0 (p 0 ) are an eigenvalue and an associated eigenvector of D f (X e (p 0 ) , p 0 ).

X (t) = X 0 (p 0 ) e λ(p 0 )t . (6)
Because of the form of the solutions, the stability nature of X e (p 0 ) is expressed by the eigenvalues of the Jacobian D f (X e (p 0 ) , p 0 ). Assume the Jacobian has no purely imaginary eigenvalue (hyperbolic equilibrium). If all the eigenvalues show a strictly negative real part then the equilibrium is asymptotically stable; if at least one eigenvalue shows a strictly positive real part then the equilibrium is unstable. The imaginary part indicates the oscillatory or non-oscillatory nature of the corresponding mode.

In the case the Jacobian is not diagonalisable, the solutions have the more general form expressed in Eq. 7, where P is a polynomial whose order is strictly lower than the multiplicity order of the associated eigenvalue λ. X (t) = P (t, p 0 ) e λ(p 0 )t . [START_REF] O'reilly | Some aspects of destabilization in reversible dynamical systems with application to follower forces[END_REF] This situation can only lead to instability if the real part of at least one eigenvalue is strictly positive or if the Jacobian has at least one purely imaginary eigenvalue. The existence of purely imaginary eigenvalues lacks of robustness with regard to physical parameters. However, the sign of the real part of the eigenvalues is a robust criterion. Furthermore, because of the continuity of the eigenvalues with regard to the elements of D f (X e (p 0 ) , p 0 ), the locus of purely imaginary eigenvalues in the parameters space form continuous surfaces, provided the equilibrium evolves continuously with p. These surfaces separate regions of asymptotic stability and instability, thus they constitute stability frontiers.

As reported in many previous works the damping structure has an important influence on the stability frontiers. In [START_REF] Kirillov | Destabilization paradox due to breaking the Hamiltonian and reversible symmetry[END_REF], Kirillov considers 2-DOF purely circulatory and purely gyroscopic undamped systems. He demonstrates for both these systems that the limit of the domain of asymptotic stability as the amplitude of a perturbing action vanishes can differ from the domain of marginal stability of the unperturbed systems, this being referred as the destabilization paradox. As a consequence, significant differences can be found between experienced and theoretically predicted stability frontiers. This effect emphasizes the strong influence of the structure of the system on the stability frontiers independently from the amplitude of the parameters.

Nature of the Hopf bifurcation

Assume the system described by Eq. 4 shows a Hopf bifurcation for p = p 0 + ∆p with regard to the element p i of p. By this is meant that f is assumed to be regular with regard to X and p i at (X e (p 0 ) , p 0 ), D f (X e (p 0 ) , p 0 ) shows one pair of purely imaginary eigenvalues λ 1,2 (p 0 ) = ±iω 0 with ω 0 > 0, all the other eigenvalues show a strictly negative real part and

d Re (λ 1,2 ) d p i (p 0 ) = 0. (8)
This situation corresponds to the crossing of the stability frontier and the transition between an asymptotically stable and an unstable nature of the equilibrium. Without loss of generality, one can assume p 0 = 0 and X e (p) = 0. For |p i | sufficiently small, let X 0 (p) be associated to λ 1 (p) and X * 0 (p) be its adjoint eigenvector, i.e. D f (X e (p) , p) X

0 (p) = λ 1 (p) X 0 (p) and D f (X e (p) , p) T X * 0 (p) = λ 1 (p) X * 0 (p), such that X 0 (p) * , X 0 (p) = 1 where •, • is the inner product [u i ] , [v i ] = i ūi v i .
According to the Shoshitaishvili theorem the topology of Eq. 4 in the vicinity of p 0 can be judged from its restriction to a local center manifold, expressed as

X (t) = ς (t) X 0 (p) + ς (t) X 0 (p) + 3 i+j=2 1 i!j! a ij (p) ς i ςj + O |ς| 4 , (9)
which is locally topologically equivalent to the restriction

X (t) = q (t) X 0 (p) + q (t) X 0 (p) , with q (t) = X * 0 (p) , X (t) . (10)
Eq. 4 can be changed into Eq. 11,

Ẋ = D f (X e (p) , p) X + g (X, p) , with g = O X 2 . ( 11
)
Then, it follows from Eq. 10 that q = λ 1 (p) q + X * 0 (p) , g qX 0 (p) + qX 0 (p) , p .

Provided f is regular enough, a polynomial expansion can be proposed, see Eq. 13,

q = λ 1 (p) q + 3 i+j=2 1 i!j! b ij (p) q i qj + O |q| 4 , (13)
with

b ij (p) = ∂ i+j ∂q i ∂ qj X * 0 (p) , g qX 0 (p) + qX 0 (p) , p (q = 0) . ( 14
)
Then, the Poincaré normal form Eq. 16 is obtained by considering a pertinent change of variable in the form of Eq. 15,

q = r + 3 i+j=2 1 i!j! h ij (p) r i rj , ( 15
) ṙ = λ 1 (p) r + γ (p) |r| 2 r + O |r| 4 , (16)
γ (p) = 2λ 1 (p) + λ 1 (p) 2 |λ 1 (p)| 2 b 11 b 20 + |b 11 | 2 λ 1 (p) + |b 02 | 2 2 2λ 1 (p) -λ 1 (p) + b 21 2 . ( 17
)
Finally, a change of variable and time scale expressed in Eq. 18 allows writing Eq. 19 for |p i | sufficiently small,

r = Re (γ (p)) Im (λ 1 (p)) - Re (λ 1 (p)) Im (γ (p)) Im 2 (λ 1 (p)) -1 / 2 s and d τ = 1 ω 0 1 -Im γ (p) ω 0 |r| 2 d t, (18)
d s d τ = Re (λ 1 (p)) Im (λ 1 (p)) + i s + sign (Re (γ (0))) |s| 2 s + O |s| 4 . ( 19
)
According to the center manifold theory, the dynamical behaviour of Eq. 4 is directly related to the behaviour of Eq. 20 in the vicinity of the Hopf bifurcation,

d s d τ = Re (λ 1 (p)) Im (λ 1 (p)) + i s + sign(l 1 ) |s| 2 s, (20)
l 1 = Re (ib 11 b 20 + ω 0 b 21 ) . ( 21
)
l 1 defined in Eq. 21 is the first Lyapunov coefficient. Assume l 1 = 0; it can be seen from Eq. 20 that a limit cycle branch connects with the equilibrium branch at the Hopf bifurcation point. This is why the Hopf bifurcation is an appropriate candidate phenomenon to explain the rise of self-generated vibrations in dynamical systems. The stable or unstable nature of this limit cycle, related to the supercritical or subcritical nature of the bifurcation respectively, can be judged from the sign of l 1 . If l 1 < 0 (respectively l 1 > 0) then the bifurcation is supercritical (respectively subcritical). In the case of a degenerated Hopf bifurcation (l 1 = 0) further developments can be proposed to determine the next Lyapunov coefficients. The interested reader is referred to [START_REF] Sotomayor | Bifurcation analysis of the Watt governor system[END_REF] and [START_REF] Takens | Unfolding of certain singularities of vector fields: Generalized Hopf bifurcations[END_REF].

The determination of the nature of a Hopf bifurcation is of particular importance when dealing with the suppression of self-generated vibrations caused by such a mechanism. Indeed, a supercritical bifurcation allows reducing the amplitude of the limit cycle as much as desired in the vicinity of the stability frontier. Therefore, an acceptable level can be reached without stabilizing the equilibrium. Contrariwise, a subcritical bifurcation leads to a jump phenomenon at the stability frontier between the equilibrium branch and the attracting steady state. Moreover, stable vibrations can remain even after the equilibrium is stabilized. Assume a polynomial expression of g in Eq. 11 in the form of Eq. 22,

g (X, p) = 1 2 P (X, X, p) + 1 6 Q (X, X, X, p) + O |X| 4 , (22)
with

P (U, V, p) = i,j 1 ∂ 2 g (X, p) ∂x i ∂x j u i v j (X = 0) (23)
and

Q (U, V, W, p) = i,j,k 1 ∂ 3 g (X, p) ∂x i ∂x j ∂x k u i v j w k (X = 0) (24)
where x i is the i-th component of X. Because of the relation between Eq. 4 and Eq. 16 above mentioned, vectors a ij (p) can be directly identified in Eq. 9 so that

ς = λ 1 (p) ς + 1 2 Γ (p) |ς| 2 ς + O |ς| 4 . ( 25
)
Writing Γ, X 0 , P (U, V), Q (U, V, W) and D f for Γ (0), X 0 (0), P (U, V, 0), Q (U, V, W, 0) and D f (0, 0) respectively, substituting Eq. 9 into Eq. 4 and identifying the vectors a ij (p) in order to fit the form of Eq. 25 for p = 0 one obtains the following criterion for the nature of the Hopf bifurcation,

L 1 = Re (Γ) , (26)
Γ = X * 0 , Q X 0 , X 0 , X 0 + P X 0 , (2iω 0 I -Df) -1 P (X 0 , X 0 ) -2P X 0 , Df -1 P X 0 , X 0 , ( 27
)
where I is the identity matrix. Of course, the criterion on the sign of L 1 defined in Eq. 26 and the sign of l 1 defined in Eq. 21 are equivalent, but the latter expression can be more convenient when an expression of g in the form of Eq. 22 is available.

Stability analysis of the squealing clutch

A complete parametric analysis of the modes and the stability domain of the equilibrium was proposed by the authors in [START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF]. Eq. 3 can be changed to fit the form of Eq. 4 by considering the state variable X = θ φ θ φ T . Provided no degree of freedom is over-damped, the characteristic polynomial of the Eq. 3 is strictly positive on the real axis. Therefore, the eigenvalues of the system can not be real. Thus, the trivial equilibrium can only destabilize by flutter. In order to distinguish the effect of the damping amount and its distribution over the two coupled degrees of freedom, one calls ξ t = ξ (1 + αβ) the total damping amount. Then, for ξ t = 0 the stability frontier is described by (see [START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF])

ϕρ + αξ t α + β 1 + αβ 2 -ξ t ϕρ + αξ t α + β 1 + αβ 1 + α 2 + ρ 2 + 4 αβ (1 + αβ) 2 ξ 2 t +ξ 2 t α 2 + ϕ 2 = 0, (28)
The frontiers for α = 1.5 are depicted on Fig. 2 and 3 for various iso-distributed damping amounts and for various damping distributions respectively. The domain of marginal stability of the associated undamped system has also been reported. On this figure, stability is found in the central area and the equilibrium experiences a Hopf bifurcation when the parameters cross the frontiers. It can be shown that the illustrated behaviour is not qualitatively affected by the value neither of α nor ξ t provided α = 1, ξ t = 0 and ξ t sufficiently small. For α = 1 the domain of marginal stability of the undamped purely circulatory system collapses and this also affects the frontiers of the damped system. However, this situation physically corresponds to the equality of the frequencies of two distinct modes and is unlikely to occur in a real system including some geometrical defects. Therefore Fig. 2 and3 correctly illustrate the respective influence of the parameters in the general case. An important observation concerns the distinct role of the damping amount and its distribution respectively. On the one hand, the presence of damping adds robustness to stability by changing the domain of marginal stability into a domain of asymptotic stability whose size depends on the damping amount. On the other hand, the damping structure rules the shape of the stability frontiers. This contrast can be emphasized in the manner of Kirillov [9] by considering a vanishing perturbation of an undamped system. As an example, assume a constant structure of the damping matrix with a fixed value of β and ρ = ηξ t , i.e. a perturbed undamped purely circulatory system. Then, as ξ t vanishes, Eq. 28 tends to

(1 + αβ) 2 1 + η 2 ϕ 2 + α 2 -1 1 -α 2 β 2 ηϕ -αβ α 2 -1 2 = 0. ( 29
)
The limits of the stability frontiers as the perturbation vanishes are reported on Fig. 4. It is obvious from this expression that the stability frontiers of the vanishingly damped system do not necessarily tend to

|ϕ max | = α 2 -1 2 , with ϕ max η = 1 -α 2 2 1 -αβ 1 + αβ . ( 30
)
This result indicates that the limit of the stability frontier when the perturbation vanishes is at most the limit of marginal stability of the unperturbed system (coalescence point), in accordance with [START_REF] Kirillov | Destabilization paradox due to breaking the Hamiltonian and reversible symmetry[END_REF]. This also reveals that the smaller the coefficients of the damping matrix are, the more important is to know the structure of this matrix in order to correctly predict the stability nature of the equilibrium, which constitutes the destabilization paradox. This makes the understanding of poorly damped systems a difficult matter. As illustrated on Fig. 4, the locus of the extrema of |ϕ| as ξ t vanishes correspond to two distinct curves described by the second part of Eq. 30. Indeed, a maximum of |ϕ| exists for both positive and negative values. These curves cross each other at η = 0 and αβ = 1. At this point both limits reach their maximum and therefore the overall stability region is the widest. A maximum width of the stability domain is also observed close to the purely circulatory system with an iso-distribution of damping for a non-vanishing damping on Fig. 3. As a matter of fact, the width ∆ϕ of the stability domain can be calculated from Eq. 28 and is a C 1 even function of ρ, which implicates that an extremum of ∆ϕ always exists at ρ = 0. The authors have reported the optimal damping distribution 10 which maximize ∆ϕ for ρ = 0 in [START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF],

(αβ) opt = 16 (α 4 -α 2 + 1) ξ 4 t + 4 (α 2 + 1) (α 2 -1) 2 ξ 2 t + (α 2 -1) 4 -4 α 2 -1 ξ 2 t 4ξ 2 t + (α 2 -1) 2 , ( 31
)
which actually tends to (αβ) opt = 1 as ξ t vanishes. Therefore, the iso-distribution of damping can be considered as a "practical" criterion of optimization of the stability domain for quasi-purely circulatory systems. It can be observed on Fig. 2 that the local extreme values of the circulatory action on the frontier evolve very slowly in the vicinity of purely circulatory systems as the damping amount increases. This emphasizes the important role of the damping structure relatively to the damping amount.

In order to complete this overview of the stability domain, the nature of the Hopf bifurcation has been investigated from Eq. 1 and reported on Fig. 5 for two different amounts of damping. It can be observed on this figure that the damping distribution strongly affects the nature of the bifurcation. Contrariwise, the amount of damping has a weak influence. Once again, a dominant role of the damping structure is suggested by this result.

Non-linear analysis and limit cycles

Although the stability analysis is a powerful tool in order to highlight phenomenological roots of selfgenerated vibrations, the prediction of their properties definitely requires taking into account the nonlinearities of the system. To this purpose, a method for determining the limit cycles of autonomous nonlinear systems is introduced in the first part. Then, a stability analysis of these periodic solutions is proposed. This last step aims at predicting if the system is likely to converge to one of these solutions or if a more complex behaviour is to be expected. Finally, the limit cycles of the example model are investigated.

Determination of the limit cycles

A method is proposed in this section to identify the limit cycle arising from the Hopf bifurcation. This method is based on an extension of the harmonic balance method. It includes dynamic constraints in order to avoid convergence of the estimated solution to the equilibrium. It shares some similarities with the Constrained Harmonic Balance Method (CHBM) method proposed by Coudeyras et al. [START_REF] Coudeyras | A new treatment for predicting the selfexcited vibrations of nonlinear systems with frictional interfaces: the Constrained Harmonic Balance Method, with application to disc brake squeal[END_REF]. Assume a system described by

Ÿ = h Y, Ẏ, p , ( 32
)
where h is a real function globally continuous with regard to both the real vectors Y and Ẏ, and a continuous and piecewise C 1 periodic solution exists. This solution can be expressed as a Fourier series,

Y (t) = A 0 (p) + n 1 (A n (p) cos (nω (p) t) + B n (p) sin (nω (p) t)), (33)
where A n (p) and B n (p) are constant vectors and ω (p) is the angular frequency of the solution. In the previous expression, ω (p) is unknown since it is likely to differ from the angular frequency of the unstable mode obtained by the eigenvalues analysis of the linearized system. The method consists in changing the time integration problem into an optimization problem. Consider the following functional,

Ψ ([A n ] , [B n ] , ω) = t 0 +2π/ω t 0 Ÿ (t) -h Y (t) , Ẏ (t) , p 2 d t. (34)
A solution of Eq. 32 is a global minimum of Ψ. In order to numerically identify it, the Fourier series is truncated at a high enough order. By doing so, the functional is changed into a function of a finite number of variables. Finally, convergence to some irrelevant local minima (as equilibrium that is an exact solution of the nonlinear expression of Eq. 32) can be avoided if the following constraints are imposed:

t 0 +2π/ω t 0 Ÿ (t) -h Y (t) , Ẏ (t) , p d t = 0, (35)
t 0 +2π/ω(p) t 0 Ÿi (t) d t - t+2π/ω t 0 h i Y (t) , Ẏ (t) , p d t = 0, ∀i, (36)
t 0 +2π/ω(p) t 0 Ÿi (t) + h i Y (t) , Ẏ (t) , p Ẏi (t) Ẏi (t) 2 + Ÿi (t) 2 d τ = 0, ∀i, (37)
where the subscript i indicates the i-th component of a vector. Theses proposed constraints correspond to a null mean error, to a mean respect of the fundamental principle of dynamics on each DOF and to a motion being neither dilating nor contracting in the phase plane of each DOF respectively. Other constraints can be used to improve the convergence, provided they correspond to some properties fulfilled by both the solution and its approximations. By solving this problem, not only one finds an approximate form of the solution but also the angular frequency is identified. Moreover, this approach allows obtaining estimations faster than a direct time integration of Eq. 32, unstable limit cycles can be identified as well, and no post-processing is required.

The method is particularly efficient with parametric studies of continuous limit cycle branches when one iteratively uses a result as an initial guess for a close new set of parameters.

Stability analysis of the limit cycles

Assume a periodic solution X of Eq. 4 has been identified having the angular frequency ω = 2π/T with T its period. The analysis of its stability aims at determining if this solution is attractive or not, i.e. if the system is likely to converge or not to this limit cycle. To this purpose, a vanishing perturbation ε of X is considered and X = X + ε denotes the perturbed solution. Then, one considers the following local approximation of Eq. 4 in the vicinity of

X ε ≈ D f (X, p) ε, (38)
which is a periodic non-autonomous but linear differential equation. Therefore, its solutions form a vector space. Consider a basis of solution {e i }. Because of the periodicity of Eq. 38, one can express the elements of this base as linear combinations of the same elements translated of one period in time,

e i (t + T ) = j 1 c ij e j (t), ∀i, ∀t or E (t + T ) = ME (t) , ∀t with E = [e i ] . (39)
The constant matrix M is the monodromy matrix. The basis {e i } can be chosen so that

E (t 0 ) = I, ( 40
)
where I is the identity matrix and t 0 is an arbitrary origin of time. It follows that

M = E (t 0 + T ) . ( 41
)
The eigenelements of M characterize the modes of perturbation of X. An eigenvector V i associated to the eigenvalueµ i is such that

MV i = µ i V i , i.e. V i (t + T ) = µ i V i (t)
. Therefore, according to the Floquet theory

V i (t) = Θ i (t) e i arg(µ i ) T t e log(|µ i |) T t , (42)
where Θ i (t) is a T -periodic function. It follows that the studied limit cycle is unstable if at least one eigenvalue of M has a modulus strictly bigger than 1. Because Eq.4 is autonomous, any translation in time of X produces a new solution. The difference between this new solution and X corresponds to a periodic perturbation. Therefore, 1 is always an eigenvalue of M. Assume one of the other eigenvalue crosses the unit circle and the remaining ones remain inside. Then, it can be shown from the center manifold theory that the crossing through 1 corresponds to the junction of a stable limit cycle with an unstable limit cycle which both disappear. A crossing through -1 corresponds to the destabilization of the limit cycle and the appearance of period-doubling bifurcation. Finally, a crossing by a pair of non-real conjugate eigenvalue corresponds to the junction of a stable limit cycle with a stable biperiodic trajectory (Neimark-Sacker bifurcation). It can be observed that this stability analysis of the limit cycle is similar to the stability analysis of equilibriums introduced in the previous section and can even be considered as its generalization. Indeed, equilibrium can be considered as a periodic solution with any value for T and the same approach can be used to define the stability. This consideration leads to

M = e D f (Xe(p 0 ),p 0 )T . ( 43
)
A translation in time of equilibrium induces no difference so there is no reason for M to admit 1 as an eigenvalue, except on the stability frontier. According to the center manifold theory, the period of the destabilizing mode and the period of the limit cycle arising from the Hopf bifurcation converge to each other at the bifurcation point. Therefore, if T is chosen so that it is equal to this common period at the bifurcation point, the exponential of the eigenvalues of D f (X e (p 0 ) , p 0 ) time T can be expected to continuously connect with the eigenvalues of the monodromy matrix of the limit cycle at the bifurcation point.

Parametric investigation of the limit cycles of the squealing clutch

The proposed method is applied at order 9 (truncation order of the Fourier series) that allows a high enough precision over the domain of investigation, and the results are compared to direct time integration on Fig. 6 for ξ t = 20%, αβ = 1, ρ = 0 and ϕ = 1. It is observed on the whole results that the qualitative results are rather independent from the damping amount. ξ t = 20% is a high value of damping. However, high values are observed to allow obtaining smoother evolutions of the limit cycles. Therefore, this example is more appropriate for preliminary observations. More realistic damping amounts are investigated later. Both the trajectory of the identified limit cycle in the state space and the estimated angular frequency are in very good accordance with the result from time integration. It can be noted in this example that the angular frequency of the limit cycle is ω = 3.85 rad.s -1 (see Fig. 5(b)) which greatly differs from the angular frequency of the unstable linear mode ω 0 = 1.31 rad.s -1 obtained by considering the stability analysis of the linearized system. Following Eq. 38, a local approximation of Eq. 1 in the vicinity ε = ε 1 ε 2 T of a limit cycle X = θ φ T can be expressed as

ε1 ε2 + 2ξ ρ -ρ 2αβξ ε1 ε2 + 1 ϕ -ϕ α 2 ε 1 ε 2 + 3σ θ 2 ε 1 φ 2 ε 2 = 0 0 . ( 44
)
The eigenvalues of the monodromy matrix obtained for the example by integrating Eq. 44 (see Fig. 7(c) and 7(d)) are inside the unit circle, which confirms the stable nature of the identified limit cycle.

The limit cycles can be repeatedly estimated in the same manner in order to perform a parametric investigation. Such a parametric investigation is reported on Fig. 7 for a variation of the circulatory action and on Fig. 8 for a variation of the gyroscopic action for the same values of the other parameters as in the previous example. The mean inclination indicated on these figures is the mean inclination of the disc over one period of the identified limit cycles as expressed in Eq. 45. This expression correspond to the instantaneous global inclination of the disc (combination of the two components) averaged over one period of the periodic motion. The associated stability nature corresponds either to stability if no eigenvalues of the monodromy matrix has a modulus bigger than 1, instability with a dominant real eigenvalue if the biggest eigenvalues in modulus is real or instability with a pair of dominant conjugate complex eigenvalue if the

ϑ = ω 2π τ =τ 0 + 2π ω τ =τ 0 θ 2 + φ 2 d τ . ( 45
)
Regarding the evolution of the limit cycle versus the circulatory action for the purely circulatory system, see Fig. 7, one can observe a unique limit cycle arising at the Hopf bifurcation with growing amplitude and frequency. The eigenvalues of the monodromy matrix of the limit cycle continuously connect with those of the equilibrium at the bifurcation point as expected. Then, one eigenvalue is equal to 1 and the other ones remain inside the unit circle, which indicates a stable limit cycle on the studied domain.

From the end of this domain (ϕ = 1) and by varying the gyroscopic action in the vicinity of 0, see Fig. 8, one can observe an abrupt evolution of both the amplitude and the frequency, and the appearance of a new limit cycle branch for ρ > 0.03. The eigenvalues of the monodromy matrix indicate that the first branch remains always stable whereas the new one shows instability by real eigenvalues for the highest amplitude part and instability by complex eigenvalues then stability for the lowest amplitude part. These figures illustrate a strong influence of the gyroscopic action on the existence of limit cycle branches and their evolution, especially in the vicinity of the purely circulatory system for an iso-distribution of damping. Indeed, a low gyroscopic action drastically modifies the limit cycle. Finally, it is important to observe on Fig. 8 that the frequency can either increase or decrease as the amplitude grows. Thus the frequency and the amplitude of the limit cycles are rather independent and show no simple relation. Fig. 9-26 allow extending these observations to various values of damping amount, damping distribution, gyroscopic and circulatory actions (the equilibrium is not reported anymore on these figures). These figures show sections of the parameters space in the same manner as on Fig. 7-8 for ξ t = 5%, ξ t = 10% and ξ t = 20%. It can be firstly checked that the subcritical or supercritical nature of the Hopf bifurcation observed on all these figures is in perfect accordance with Fig. 5.

The limit cycle branches and their stability domains tend to be more separated in the parameters space as the damping amount increases. However, no important qualitative effect is observed on their shape and relative positions. The most important effect of the damping amount is quantitative: the more damping the lower amplitude and frequency of the limit cycles. The influence of the circulatory action has also a general tendency, in spite of local discrepancies on unstable limit cycles: the more circulatory action, the larger amplitude and frequency. The fast evolution of the limit cycle with regard to the gyroscopic action in the vicinity of the purely circulatory system with an iso-distributed damping which was previously mentioned can be generalized. Indeed, it can be observed for any damping amount and an iso-distribution of damping that when the purely circulatory system shows an unstable equilibrium, a change in the gyroscopic action leads to a fast evolution of the limit cycle, see Fig. 19, 22 and 25. For negative values of the gyroscopic action, a unique limit cycle with a large amplitude and frequency is observed. This limit cycle evolves slowly for large (negative) values of the gyroscopic action. For positive values of the gyroscopic action, the amplitude rapidly decreases, reaches a minimum then increases slowly. For a large enough gyroscopic action, a new limit cycle branch can be observed with the same qualitative behaviour as previously discussed. Fig. 22 suggests that the highest amplitude part and the lowest amplitude part of this new branch are likely to connect to form a closed branch for a large enough gyroscopic action. The highest amplitude part of this new branch is unstable by a dominant real eigenvalue of the monodromy matrix whereas the lowest amplitude part is firstly unstable by complex eigenvalues then becomes stable and its amplitude and frequency are close to those of the first branch over a range of the gyroscopic action. For a lower circulatory action, i.e. when the purely circulatory system with an iso-distributed damping has a stable equilibrium, the first branch gets separated into two parts which surround the stable equilibrium. In this situation a negative gyroscopic action leads to a subcritical bifurcation with an important Figure 26: Limit cycles versus the gyroscopic action for various values of the circulatory action for Eq. 1 with α = 1.5, αβ = 5, ξ t = 20% and σ = 10 4 , (a) mean inclination, (b) Relative frequency (gray: stable limit cycle, dot black: dominant real unstable eigenvalue, solid black: dominant pair of conjugate complex unstable eigenvalues) jump phenomena whereas a positive gyroscopic action leads to a supercritical bifurcation. The additional branch appears for a large enough circulatory action. The fast evolution of the limit cycle which is observed in the vicinity of the purely circulatory system for αβ = 1 is shifted to negative values of the gyroscopic action as αβ is shifted to αβ < 1. As a consequence, the purely circulatory system shows a far lower limit cycle amplitude and frequency and a very low sensitivity to the gyroscopic action, see Fig. 18, 21 and 24. The possibility to reach a minimum of amplitude and minimum of sensitivity with regard to the gyroscopic action by varying the damping distribution is suggested by this result. In the meanwhile, the additional limit cycle branch is shifted to positive values of the gyroscopic action and positive values of the circulatory action. The amplitude and frequency of its lowest amplitude part gets separated from the first branch and their growth is slower with regard to both the circulatory and the gyroscopic actions, whereas the first branch shows a faster growth. Nevertheless, apart from these points and the modification of the stability frontier of the equilibrium, the limit cycle branches look very similar for αβ = 1 and αβ < 1. Contrariwise, the behaviour of the limit cycle branches becomes more complex as αβ is shifted to αβ > 1. Fig. 20, 23 and 26 reveal that all the parts of the branches have changed their connections in the vicinity of the purely circulatory system and now all branches interconnect. On Fig. 20, 23 and 26, it can be observed that the limit cycles are unstable by complex eigenvalues of the monodromy matrix where the parts connect. Moreover, a particular effect can be observed on Fig. 20, 23 and 26 for the purely circulatory system which now shows a subcritical bifurcation. A subcritical bifurcation with no stable limit cycle is even observed for the lowest values of the damping amount, see Fig. 11 and14. As a consequence when the circulatory action crosses the stability frontier a jump phenomenon to a steady state which is not a limit cycle is obtained. Another consequence of the interconnection between the limit cycle branches is that the steady state of the system can depend on the time history of its parameters, independently from considerations about initial conditions. Indeed, assume the dynamics of the system remains always close to the identified limit cycles for slowly varying parameters and follows their branches by continuity. As an example, consider Fig. 17 and 26. Assume a negative gyroscopic action, e.g. ρ = -1, and a slowly increasing circulatory action, from ϕ = 0 to ϕ = 1. The system shows a unique stable limit cycle as the equilibrium destabilizes. Moreover, no other stable solution could be identified, thus the steady state is not conditioned by the initial conditions. Then, assume ϕ is kept constant and ρ varies from ρ = -1 to ρ = 1. Then, the arrival state is different of that reached when varying ϕ from ϕ = 0 to ϕ = 1 with ρ = 1 kept constant. A jump phenomena between limit cycles corresponding to a change of branch can even be expected in the later case if ρ is then decreased from ρ = 1 to ρ = -1.

To summarize, some general tendencies can be extracted from this parametric study on the role of the damping structure, the coupling actions and the nonlinearity.

• The stiffening rate of the system for non-vanishing motions (σ in Eq. 1) has an independent and strong influence as previously mentioned, the more stiffening the lower amplitude but without any effect on the frequency. • The damping amount has a rather independent effect too, the more damping the lower amplitude and frequency. • The circulatory action shows a general tendency to increase both the amplitude and the frequency.

• The non-symmetrical dependency of the limit cycle to either positive or negative gyroscopic action or damping distribution with αβ < 1 or αβ > 1 was highlighted. • For the purely circulatory system with an iso-distribution of damping, a high sensitivity to the gyroscopic action is observed. Changing the damping distribution to either αβ < 1 or αβ > 1 alters this phenomenon by shifting the fast evolution front to negative values of the gyroscopic action so that the amplitude and frequency can be largely reduced for the purely circulatory system. Figure 27: Limit cycle of the purely circulatory system versus the damping distribution and the circulatory action for Eq. 1 with α = 1.5, ξ t = 20% and σ = 10 4 (black: stable limit cycle, gray: dominant real unstable eigenvalue, white: dominant pair of conjugate complex unstable eigenvalues)

It was also conjectured the possibility to reach a robust minimum of amplitude and frequency by varying the damping distribution for the purely circulatory system. As a matter of fact, the iso-distribution of damping seems to maximize the amplitude and frequency. This particular property is emphasized on Fig. 27 which shows the evolution of the unique limit cycle identified for the purely circulatory system over a range of damping distribution. It appears clearly on this figure that the iso-distribution leads to a faster increase of the amplitude and frequency with regard to the circulatory action and represents a local maximum far from the stability frontier. Contrariwise, local minima can be observed for either αβ < 1 or αβ > 1. The bifurcation nature changes from supercritical to subcritical as αβ changes from αβ < 1 to αβ > 1. Thus not only avoiding an iso-distribution of damping can allow reducing the limit cycle amplitude but αβ < 1 also appears more favourable than αβ > 1 because of the nature of the associated bifurcation. As an important consequence, a proper design strategy can allow reducing the amplitude of self-generated vibrations and avoiding subcritical bifurcations in the same time by adjusting the damping distribution. It was shown in the stability analysis of the equilibrium that the widest stability domain is found near the iso-distribution of damping (αβ = 1). Therefore, in a small enough vicinity of the destabilization point, the iso-distribution of damping can appear as a local minimum of amplitude because of the late rise of the limit cycle. However, this situation reverses far from the destabilization point and the iso-distribution of damping becomes a local maximum of the amplitude. In order to explain this observation, it can be reminded that for the purely circulatory system with an iso-distribution of damping, both stable and unstable modes have the same frequency (see [START_REF] Hervé | Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects[END_REF]). Thus, an intrinsic resonance of the stable mode can be evocated to explain this particular behaviour. Therefore, the iso-distribution of damping restrains the most the purely circulatory system from destabilizing but to the detriment of the amplitude of the dynamic state that follows the destabilization. This can be considered as a dynamic extension of the destabilization paradox. As a concluding remark, the important contrast between the results from the stability analysis of the equilibrium and the limit cycle amplitude regarding the influence of the parameters highlights the large difference which exists between a linear and a nonlinear investigation of the system. These two viewpoints are complementary and require an equal attention in order to define relevant design strategies. Countermeasures to self-generated vibrations based exclusively on a stability analysis of the equilibrium would suggest an iso-distribution of damping which can lead to even higher vibrations amplitude. Contrariwise, taking into account the limit cycles investigation allows defining two distinct approaches depending on whether or not the instability can be retained.

Some comments on the nonperiodic dynamic behavior

In some circumstances neither a stable equilibrium nor a stable limit cycle can be found, as an example see Fig. 23 for ϕ = 1 and ρ ∈ [-0.01, -0.05]. This situation can not be highlighted by the sole identification of the limit cycle with the proposed method. The stability analysis of the estimated solution is also necessary. Indeed, a multi-DOF second order system can admit steady solutions which are neither equilibrium nor limit cycles, but dense trajectories in the state space, strange attractors or chaotic motions.

In the present section, it is proposed to illustrate such dynamic behaviours for the proposed system. A practical way to do so is to use Poincaré sections, see [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields[END_REF].

In the state space, assume a lower dimensional subspace crossed by the studied trajectory, called a Poincaré section. Then, the locus of the recursive crossings of this subspace by the trajectory in a chosen direction forms the Poincaré application and is intimately related to the nature of the motion. Such sections are reported on Fig. 28 for various values of the gyroscopic action and the same values of the other parameters as on Fig. 23. Depending on the gyroscopic action, the Poincaré section shows various pattern.

• For ρ = 0, the Poincaré section is restricted to a unique point, which indicates a periodic motion, i.e. a limit cycle, confirmed by the time evolution of the variables. • For ρ = -0.01, the Poincaré section is a single loop closed curve, corresponding to the section of a torus in the state space, thus the motion is biperiodic. The time evolution of the variables shows pseudo-periodic oscillations limited by a periodic envelop. Moreover, eigenvalues of the monodromy matrix of the identified limit cycle cross the unit circle through non-real values between ρ = 0 and ρ = -0.01, thus a Neimark-Sacker bifurcation, i.e. between a limit cycle and a torus, see [START_REF] Sacker | On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations[END_REF], can be conjectured to occur. • For ρ = -0.02, the previous closed curve now appears divided into two imbricated loops, which can indicate a period doubling bifurcation of the secondary period of the torus between ρ = -0.01 and ρ = -0.02. This is qualitatively supported by the previously observed envelop being changed into a half-frequency new one. • For ρ = -0.03 and ρ = -0.04, the Poincaré section show scattered points in the vicinity of the previous closed curve and the corresponding motion is rather irregular. A bifurcation to a nperiodic motion with n ≥ 3, a strange attractor or a chaotic attractor according to the Ruelle-Takens scenario can be evocated, see [START_REF] Ruelle | On the nature of Turbulence[END_REF]. • For ρ = -0.05, the Poincaré changes back to a closed curve, indicating a biperiodic motion.

• Finally, for ρ = -0.06 the Poincaré section is restricted again to a unique point, which indicates the stabilization of the limit cycle.

By comparing Fig. 23 and 28, it can be observed that the mean amplitude and the mean pseudo-period of the non-periodic motions remain close to those of the identified (unstable) limit cycle. Therefore, even if the limit cycle is not the attractive state, its characteristics are representative of the steady motion and can still serve as a design criterion to some extend. This was also observed for other branches of unstable limit cycle.

In spite of the relative simplicity of the proposed model, a wide variety of behaviours is observed when both the equilibrium and the limit cycles destabilize. Therefore, one can presume that more detailed models with less regular nonlinearities are even more likely to show such a complexity.

Conclusion

A nonlinear 2-DOF system combining damping and both circulatory and gyroscopic actions has been extensively investigated. Regarding the equilibrium, an important effect of the damping structure on both the stability frontiers and the nature of the Hopf bifurcation has been exhibited. In particular, it appears that the iso-distribution of damping is nearly an optimum for the stability range of quasi-purely circulatory systems. The influence of the gyroscopic action has been illustrated too.

The identification of the limit cycles of the systems also revealed a particular influence of the damping distribution. Both the connections between the branches of limit cycles and their stability are affected. As an extension to the destabilization paradox, the iso-distribution of damping has been shown to induce the fastest growth of the limit cycles for quasi-purely circulatory systems. As a result, the iso-distribution of damping appears to be at the same time the most desirable structure in order to avoid the destabilization and the worst configuration in the purpose of reducing the amplitude of the self-generated vibrations. An internal resonance of the stable mode to the unstable mode is evocated to explain this particular behaviour. Moreover, the damping structure as well as the sign of the gyroscopic action play an important role in the nonlinear behaviour of the system. In particular, no symmetry can be observed on the branches of limit cycles between positive and negative values of the gyroscopic action in the proposed model. The influence of the damping structure shows no symmetry neither. Finally, non-periodical stable motions have been shown to take place when neither stable equilibrium nor stable limit cycle were identified, including multi-periodic motion, strange or chaotic attractors. This large variety of behaviours exhibited for the rather simple autonomous 2-DOF lumped system allows emphasizing how complex the dynamics of real-life systems are likely to be. Although the proposed approach can be applied to some more sophisticated models, some major aspects of friction-induced selfgenerated vibrations are highlighted in this paper. Such minimalist models with few DOF can serve as effective tools for understanding the role of some physical parameters in identified coupling mechanisms.

Figure 1 :

 1 Figure 1: Model of rubbing discs subject to a flutter destabilization

8 -Figure 2 :-

 82 Figure 2: Stability frontiers for α = 1.5 and iso-distributed damping (αβ = 1), bold line: no damping, dashed line: ξ t = 5%, solid line: ξ t = 10%

Figure 3 :(Figure 4 :

 34 Figure 3: Stability frontiers for α = 1.5, and ξ t = 10% and various damping distributions

Figure 5 :

 5 Figure 5: Stability frontiers for ϕ > 0 and nature of the Hopf bifurcation versus the damping distribution and the gyroscopic factor for α = 1.5, ξ t = 5% (a) and ξ t = 20% (b) (dark: subcritical bifurcation, light: supercritical bifurcation)

  DOF (dθ/dτ) Velocity 2 nd DOF (dφ/dτ)

Figure 6 :

 6 Figure 6: Comparison between time integration and limit cycle estimation (gray: time integration, black: estimated limit cycle) of Eq. 1 with α = 1.5, αβ = 1, ξ t = 20%, ϕ = 1, ρ = 0 and σ = 10 4 , (a) trajectory (b) velocity phase

Figure 7 : 1 -Figure 8 :

 718 Figure7: Equilibrium in its stability domain (black) connecting to the arising limit cycle (gray) versus the circulatory action, for Eq. 1 with α = 1.5, αβ = 1, ξ t = 20%, ρ = 0 and σ = 10 4 , (a) mean inclination, (b) relative frequency (least stable mode and limit cycle respectively), (c) and (d) eigenvalues of the monodromy matrix and unit cylinder

Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :Figure 14 :Figure 15 :Figure 16 :Figure 17 :Figure 18 :Figure 19 :Figure 20 :Figure 21 :Figure 22 :Figure 23 :Figure 24 :Figure 25 :

 910111213141516171819202122232425 Figure9: Limit cycles versus the circulatory action for various values of the gyroscopic action, for Eq. 1 with α = 1.5, αβ = 0.2, ξ t = 5% and σ = 10 4 , (a) mean inclination, (b) Relative frequency (gray: stable limit cycle, dot black: dominant real unstable eigenvalue, solid black: dominant pair of conjugate complex unstable eigenvalues)

Figure 28 :

 28 Figure 28: Steady solution of Eq. 1 with α = 1.5, αβ = 5, ξ t = 10%, ϕ = 1, σ = 10 4 and various values of ρ, (a) Poincaré application in the section θ = 0 and θ > 0, (b) position on the 1st DOF versus time 39