Fabrice Flahutez

Roberto Matta

Yves Tanguy

Kay Sage

Richard B Freeman

Trained as an architect, Roberto Matta would focus on science in general, taking concepts as well as pictures from books of popular science. When Matta

original vocabulary of forms with roots in the infinitely small and the cosmic immensity. "Gordon Onslow Ford says that [...] Matta, his eyes glued to a microscope, focus on flowers, insects, snail shells" 5 . There is no question of a Newtonian physics as Matta or Breton would have dreamed, but a science derived from Einstein or Riemann. This fascination with Einstein is recorded in the pages of one of his scrap books illustrated with several photographs of the German scientist. Such an attraction is understandable because the physics of relativity is the little sister of intuition and analogy. This dual interest in that which constitutes us (cells, molecules) and that which contains us (stars, nebulae, space-time) paves the way for an inventive painting which resonates with Andre Breton's purely intellectual interest in the esoteric and the occult. What is the connection between these strange worlds? What are the terms of convergence of such diverse thoughts? Roberto Matta's painting is at the center of the transformations of Surrealism during the war until the birth of the "Prolegomena to a third Surrealist Manifesto or not," 6 and the Grands Transparents. 5 Paul Haim : Matta, agiter l'oeil avant de voir, errances, souvenirs et autres divagations, Éditions Séguier, Paris, 2001, p. 33. 6 André Breton : OEuvres complètes, t. III, bibliothèque de la Pléiade, Gallimard, Paris, 1999, pp. 5-15. This text is written in 1942 in New York.

During the years 1937-1939, the subjects are biomorphic and are very close to structures of eukaryotic cells 7 . In the center of the colored circular shapes there are networks of lines that resemble colored representations of the Golgi apparatus 8 or mitochondria 9 found in scientific books.

In the drawing Untitled 10 of 1937, forms interpenetrate where the colored shapes suggest several internal organelles or some metabolic galleries and microscopic organisms. These forms are also present in the details of X-Space and the Ego in 1945.

Cell organelles indicate the presence of life and a growing world. 7 Fabrice Flahutez : La peinture de Roberto Matta entre cellule eucaryote et singularité de Schwarzschild, in : Mélusine, (le surréalisme et la science), Cahiers du Centre de Recherche sur le Surréalisme (Paris 3 Sorbonne-Nouvelle), Henri Béhar (dir.), n° XXVII, Éditions L'Age d'Homme, Lausanne, March 2007, pp. 145-153. 8 Camillo Golgi (1843-1926) was an Italian physician, pathologist, scientist, and Nobel laureate. Golgi identified the intracellular reticular apparatus in 1898 which bears his name, the Golgi apparatus. 9 In cell biology, mitochondria are membrane-enclosed organelles found in most eukaryotic cells. Mitochondria are described as cellular power plants because they generate most of the cell's supply of ATP used as a source of chemical energy. 10 Roberto Matta : Untitled, 1937, watercolor on paper, 31,1 x 49, 5 cm, The Ira and Gale Drukier Collection, USA. See Germana Ferrari, n°D37/28.

[ Fig-2 ]

Biology is also the focus of his research, an example of which can be seen in the detail of the wide-open mouth, which is the strong jaws of the lamprey for the cover of issue 4 of the magazine VVV, released in February 1944. Roberto Matta uses this marine animal because it is the only one that reverses the sexual behavior of the praying mantis 11 . This reversal is symbolic insofar as surrealism has always tried to reverse values. Rather than adopting the historic bestiary, Surrealism reinvented the vocabulary of flora and fauna through affinity and analogy. The aggressive acronym VVV highlights the idea of Surrealism's offensive in exile, and the lamprey, which characteristically attaches itself, even to stones with the help of a suction cup mouth, is the symbol of resistance to disappearance. But the lamprey is also a snakehead, a migratory vampire that feeds on its host with its fanged mouth 12 . This animal, which has both resisted any changes and remained intact despite the evolution of species, metaphorically embodies the immortality of the vampire that sucks the blood of its victims. The cover of VVV, no. 4 is entirely dark red, furthermore, the lamprey is mainly cooked in its own blood. It should be noted also that the specie used by Matta was discovered in 1911 date of his birth.

Beyond the study of the infinitely small or the most unusual bestiary, Matta also looked into the telescope and invented the word « Forms gradually become fluid, the colors make love in the sense that they blend with each other, while the angle is categorically rejected.

The world is unstable, hazy or liquid and structures appear as broken 17 windows. The round shapes are like small clouds, whose blurred edges are created by erasing the charcoal drawing and colors. These spherical objects where several détails are not precise, but fluid and blurred are related to Heisenberg's uncertainty principle, which Matta studied in 1943 (p. 214 of his scrap book from 1943). According to Heisenberg's principle, it is impossible to simultaneously measure the present position of a particle while also determining its future motion. In Matta's drawings and paintings the circular volume is bubbling because it expresses a becoming, a dynamic and psychological morphology.

The details disappear in favor of flexible, diaphanous, melting elements. In his scrapbook, he tirelessly draws many corpuscular shapes like molten metal. The molten elements are surrounded by planar geometric figures expressing levels of space. Thus, to paint the structure of space, making it visible, Roberto Matta has used parallelograms which suggest that a window has just exploded. This is the only way to make visible the transparency of glass. "The verb is glass," said Matta, "the language is clear as crystal." The compositions in "fusion", where the colors blend, are described by Gordon Onslow-Ford as the meeting point of the feminine and male principles 19 and in an other hand, there is also a kind of reconciliation between microcosm and macrocosm with the human figure in the center of reflection. The chromatic eruptions are also a metaphor for the intense bubbling of the unconscious20 and the bustle of life. Many drawings are based on this connection between feminine and male principles and/or between bubbling and the figure of motherhood.

(Roberto Matta's wife will give birth to twins). The spiral is finally the expression of the surrealist labyrinth in the centrifugal sense. Roberto Matta also uses a specific mathematical 34 The spiral appears on the cover of the journal Dyn, no. Ellimonde [START_REF] Matta | Ellminonde[END_REF] , Disasters of Mysticism [START_REF] Matta | Disasters of Mysticism[END_REF] and Black Virtue [START_REF] Matta | Black Virtue[END_REF] show these distortions of space, where the image works as a gateway to other dimensions. Paintings titled The Onyx of Electra [START_REF] Matta | The Onyx of Electra ("The electronics[END_REF] , Galaxies (Mysticism of Infinity) [START_REF] Matta | Galaxies (Mysticism of Infinity)[END_REF] , La Révolte des contraires 48 also clearly illustrate the importance of these spaces full of waves, stars, galaxies, black holes. In a drawing from an unpublished illustrated manuscript

Redefinitions Towards a New Dimension For Painting (1937-1938) exhibited at the Galerie Malingue in Paris in 2003, Matta shows how bodies can mate with these singularities that allow the meeting even if the two sides are separated by space and time [START_REF] Matta | Redefinitions Towards a New Dimension For Painting[END_REF] .

In this drawing, the male and female bodies make love without touching, the space is deformed, so that harmony is restored. The "The Hertzien oscillation is looking at you," states Matta in the picture caption . 55 The artist created an analogy between the two magnetic arcs and eyes, which gives an anthropomorphic conception of the invisible in accordance with André Breton's theory of the Grands transparents.

The "mattaïque" challenge favors a universe made of wavelengths and energy. In his paintings, the absence of objects (but the presence of body) is understood as the transformation of matter into energy. 56 The absence of objects in favor of energy is not a path to abstraction, but a way to express the essence of the world. His friend Lionel Abel recalls how the painter was fiercely opposed to the representation of the object as a subject. In an article in Commentary, he repeats the words that Matta defended in New York : "There will never be a recognizable object in a picture of mine. Never. Not one object, not the shadow of an object. No object is good enough to be in a picture of mine. No woman 55 The catalog is under the aegis of Alfred Jarry (1873-1907) : The attraction for this type of geometry 58 , the relations between art and science, his admiration for Tanguy and Duchamp, astronomy, chemistry, biology and mathematics will lead to the birth of Matta's Psychological Morphologies. [START_REF] Onslow-Ford | Note sur Matta et la peinture[END_REF] It is also the beginning of an adventure combining the aesthetic vision of the microscopic world with a macroscopic view, assimilating in one extreme condensation every

  Mattuscope » (Matta + scope) to describe this kind of machine13 . Matta's paintings then proceed by fragmentation patterns and the color becomes more varied. The psychological morphologies are a kind of culmination of drawings from 1937 to 1942. The development of psychological morphologies based on the study of Edouard Monod-Hersen 14 and Freud's theories are also developped by Matta to obtain a scholarship at the New School for Social Research in New York in 1940 (and probably funded by Peggy Guggenheim) 15 as we can read in the unpublished letters to Breton. Matta defines drawing as an experimental process, because the drawings are always the site of formal inventions which appear later in his oil paintings. The contrasts of color are violent 16 ; black and yellow, blue, pink and yellow are mixed directly.

[

  Fig-3 ] "Matta was in possession of an entirely new color range," wrote Andre Breton in Surrealism and painting, "perhaps the only, certainly the most fascinating, which has been proposed since Matisse. This new range is organized from a pink-purple in a process of transformation." 17

[ Fig- 4 ]

 4 The phrase near the drawing reproduced on p. 45 of his notebook of 1943 shows his great fascination with anything related to the cosmic: « In recent years," he writes, "a high penetration of radiation on the earth comes from outer space."

  [ Fig-5 ] In a commentary published in an article by Sidney Janis, Matta says: "creating a painting is a phenomenal experience, colossal, in the sense that man is innovative while nature invents always. It's like participating in advance in future times. […] painting the colossal structure of life, as science can geometrize the city." 21 His attraction to the cosmic spaces makes him interested to recent observations made by astronomers. In the central part of the painting ETATEAT 22 spherical white shapes float above concentric lines. The contour line combines two line segments that connect but do not intersect; these represent elevation or distortion on a topographic map. The stones seem to be absorbed by these topographic depressions as the stars are inexorably swallowed by the black hole from which even light can not escape. The composition is organized around planes which also suggest broken bidimentional spaces, like glass plates in homage to Marcel Duchamp. The Large glass by Duchamp is a multidimensional piece of art. Matta vocabulary borrows from Max Ernst, Joan Miró, Yves Tanguy for the representation of space, and from Duchamp and Giacometti for objects and themes 23 . The paintings L'année 44 24 or The Vertigo of 21 Sidney Janis : The School of Paris comes to US, in : Decision, vol. 2, no. 5-6, November-December 1941, New York, pp. 85-95. 22 Roberto Matta, ETATEAT, 1944, oil on canvas, triptych, 86 cm x 247 cm, sold : Sotheby's Paris, July 2&3 2008. 23 Rubin, William Stanley : Dada, Surrealism, and Their Heritage, Museum of Modern Art, Greenwich, Conn, distributed by New York Graphic Society, New York, 1968. p. 169. "These bizarre anthropomorphs, which became increasingly sculpturesque in their modeling as Matta proceeded, recalled the monsters of Ernst, Giacometti's Invisible Object and the aggressive mannequins in Masson's drawings of 1939-1941." Eros 25 are also attempts to express the notion of space-time as it was conveyed by American popular scientific journals 26 . [ Fig-6 ] The unknown journal Titled Instead supervised by Matta shows many scientific topics even on covers 27 . The issue n° 1 shows a meristem and the n° 7 a cloud of points and lines related to Heisenberg uncertainty principle. A meristem in biology is the tissue in plants consisting of undifferentiated cells (meristematic cells), found in zones of the plant where growth can take place. The meristematic cells give rise to various organs of the plant, and keep the plant growing.

[ Fig- 7 ]

 7 The centers of these spirals 34 are like the eye of the storm where the fusion of opposites takes place, but they are also metaphorical meeting points of Eros and Thanatos, of what Bataille called in his book "Erotism: Death and Sensuality" the passage of discontinuity to continuity35 . In addition, Matta often includes the letters OE in his titles. The body and flesh, erotism and death are embodied in these two letters found in foelie, yeux, foeu, poet.

[Fig- 9 ] 53 [

 953 concept of harmony was developped in another article about the philosopher Charles Fourier, indian conception of the universe, esoterism and surrealism 50 . In another drawing from the manuscript, genders are mixed and the result is a hermaphrodite. The drawing is formally very close to the Attalante Fugiens by Maier, one of whose engravings will illustrate the Great Transparencies of André Breton. By 1941, in the journal Decision, Matta is portrayed as a painter who is between science and art: "At the same time his penetrating visual perception X-rays reality, and by a fusion of techniques out of painting, 50 See Fabrice Flahutez : Nouveau Monde et Nouveau Mythe. Mutations du surréalisme de l'exil américain à l'Ecart absolu (1941-1965), Les presses du réel, Dijon, 2007.architecture, fumage, and photography, both images are surimposed upon the screen of his canvas where the composite becomes, in effect, an X-rayograph."51 The paintings of the war years are also particularly interesting in that they attempt to express a multi-dimensional universe. For this, the artist used fluorescent pigments that are likely to react to the wavelength of light (visible or invisible spectrum) which illuminate the painting. In a unpublished letter to Breton, Matta explained his experiments: "I dive into the black light. There are many possibilities as some colors disappear to the eye depending on the lamp used." 52 This is a completely original device to bring out images from the original pictures. Indeed, if the picture is illuminated with a black light or wood lamp, some pigments can be excited and therefore certain lines or points will react specifically to a particular light or wavelength. The transition from one wavelength to another animates the painting in which other dimensions of the image appear. The picture becomes multidimensional. On several levels, the picture vibrates under fluorescent and divergent tones that Matta will continue to use for a number of his paintings. Julien Levy explains in one of his texts that51 Matta Echauren, in : Decision, vol. 2, no. 5-6, New York, November-December 1941.52 See letter from Matta to Breton dated March 12 1940 (sent from New York to Poitiers), bibliothèque littéraire Jacques-Doucet, Paris, BRT.C. 2098. "Matta found a shop that sold fluorescent mineral paints, bought a special light to activate the combustion and with the black light to activate color mineral pigments, and began a series of paintings that used these fluorescent colors." Fig-10 ] Moreover, Matta also looked at the shapes of magnetic fields. We find these forms in the popular literature of magnetic fields, especially the work of Max Born who explains the principles of Einstein's theory of general relativity for a lay audience. 54 These representations of the magnetic fields are used by Matta and especially for the cover of the exhibition catalog of his works at the Pierre Matisse Gallery in April 1942.

  moment of perception. The first article by Gordon Onslow-Ford, which appeared in the London Bulletin, helps to link the Psychological Morphologies and The Great Transparent Ones defined by André Breton: "The psychological morphologies shape the world by our wildest thoughts. (…)There are objects that correspond to "characters", stars, trees, and architecture. (…) Objects that are only visible under a microscope." 60 Like most surrealist artists, Matta's interest in biology continues into the 1990s including the use of the pattern of radiolarians he borrows from the zoologist and illustrator Ernst Haeckel (1834-1919) and his famous book « Report on the Radiolaria Collected by the HMS Challenger During The Years 1873-1876 61 ». This loan is not a coincidence because Haeckel popularized and promoted Charles Darwin's theory in Europe and developed the controversial recapitulation theory (ontogeny recapitulates phylogeny) claiming that an individual organism's biological development (ontogeny), condense and correlate its species' evolutionary 61 Haeckel Ernst, Report on the Radiolaria Collected by H.M.S. Challenger During the Years 1873-1876, London: Eyre & Spottiswoode, 1887.

  ontogeny is linked to phylogeny because ontogeny recapitulates phylogeny. Matta looks in the microscope and in the telescope at the same time like Freud trying to connect individual psyche and the collective thought. In the aquatint etchings N'ou's T'ou's (all of us) (1995) or the painting La terre et ses oignons (1994), the sphere illustrates litterally the link between microcosm (structures and organic endoplasm as it is represented in radiolaria) and macrocosm (starsspace-time). N'ou's T'ou's is also the poetic place of a metaphoric brain where memories and dreams are working to build the origin of a new life or a new myth, remembering the all steps of humanity according to the principle of Haeckel and the Psychological morphologies by Matta. Youniverso 1998 sans titre 1999 la terre et ses oignons 1994 A letter from Roberto Matta to André Breton dated 1941 could serve as a conclusion: "[I'm] in the comparative study of time and space, I do not think the sun is so far as they claim. 71 " 71 Letter from Roberto Matta to André Breton dated 1941 from the poet collection, sold in 2003. See Andrébreton.fr, ref : 1479000. Roberto Matta : Notebook 1943, Sometimestudio, Paris, 2011, p. 45.

Fig. 5 :

 5 Fig.5 :

Fig. 6 :

 6 Fig.6 :

Fig. 7 :

 7 Fig.7 :

Fig. 8 :

 8 Fig.8 :

Fig. 9 :

 9 Fig.9 :

Fig. 10 :

 10 Fig.10 :

  11 About the praying mantis and its morals, see Roger Caillois : La Mante religieuse, de la biologie à la psychanalyse, in : Minotaure, n°5, Skira, may 1934, pp. 23-26 ; Roger Caillois : La Mante religieuse, recherche sur la nature et la signification du mythe, Aux amis des livres, Paris, 1937, and William Pressly : The Praying Mantis in Surrealist Art, in : The Art Bulletin, December 1973, pp. 600-615. 12 Claude B. Renaud : Lampreys of the world : an annotated and illustrated catalogue of lamprey species known to date, Food and Agriculture Organization of the United Nations, Rome, 2011.

  Andrea Caffi, old, (...). Regardless of (...) the scientific insights of psychological morphology, Gestalt theory 29 , astrophysics, histology and molecular physics. The need for such research does not express anything else than the desire to expand, with the help of the most modern resources, the scope of the visible."30

	Matta's	collaborator	in	the	rare	journal	Instead,
	specified: "Cosmogony, geology, the development of vegetable and
	animal species can be fitted without arbitrariness into the same series
	with the annals of all the human collectivities that have lived […] thus
	254.
	30 André Breton : Surrealism and Painting, op.cit., (see note 17).
	31 Parker Tyler : Art News, vol. 56, n° 7, November 1957, p.11.
	32 Letter from Matta to Breton dated 1944 and sent from New York to Percé in
	Canada. Breton Collection. See item n° 1157 (Breton auction April 11 & 12 2003,
	Hôtel Drouot, Paris (étude Calmels-Cohen).

to turn around the celebrated hypothesis of Laplace, it seems permissible to conceive, side by side with the omniscience condensed in a gigantic mathematical formula (positions and relations of everything that existed, exists or will exist in space-time), of a "memory" woven of nothing but qualitative moments instead of quantitative coordinates, and capable of evoking the peculiar vicissitudes of every atom, and of all the combinations of atoms, in the eternity of the universe with the contradiction, so insoluble to us, between the reality of the present and the unreality of what is no more or is not yet, as well as between the innumerable possibilities, which are not however just nothing and all that chance has let be realised."

28

If these remarks are theorized in the journal Instead in 1948, they provide a synthesis of research that Matta had recently conducted with André Breton in the U.S. Breton spoke of Matta's influences: "Thus, also, he invites us constantly to a new space, in deliberate break with

28

Andrea Caffi : History and the News Item, in : Instead, no. 1, February 1948, np. the Parker Tyler also mentioned in Art News this genesis: "Matta's early "cosmic systems" are simply astronomized mutations of Freudian visceral metaphysics, made from inspiration as passionate and immediate as a child's." 31 His paintings show spherical particles that seem to levitate in a interstellar world where the concepts of high and low, heavy and light, great and small are absent or obscured. There is a clear collusion between the old esoteric philosophies which advocated the abolition of opposites and the scientific concepts that reveal the futility of our Euclidean physics and our perception of time

32

. In his scrapbook from 1943, Matta wrote of the programmed death of geometry because it was not adapted to a global vision of the world 33 .

29

Andre Breton was very interested in Gestalt theory at the end of 1941 as evidenced by his correspondence. Letter from Andre Breton to Benjamin Peret dated January 4, 1942. (New York to Mexico), bibliothèque littéraire Jacques-Doucet, Paris, BRT.C.

33

Germana Ferrari, op. cit., (see note 2), p. 117. excerpts from Scribble-in-book, 1939-1940 and in many pages of the notebook 1943.

  6, November 1944. The spiral is directly borrowed from Theodore Andrea Cook : The Curves of life, being an account of spiral formations and their application to growth in nature, to science and to art ; with special references to the munuscripts of Leonardo Da Vinci, Constable and Company LTD, London, 1914, p. 460.35 Georges Bataille : L'Érotisme, Minuit, Paris, 1957, pp. 20-23. (Erotism: Death and Sensuality, City Lights Books, San Francisco, 1986). pattern which is the Karl Schwarzschild singularity. The inclusion of mathematical 36 and astrophysical data had started a few years earlier, when Matta (with Gordon Onslow-Ford) discovered the mathematical objects at the Institute Henri Poincaré 37 in Paris in 193738 . According to the painter, the attraction for maths is the key to the discovery of light and time.39 The spatial effects suggested there, along with ideas derived from the contour lines of the mathematical objects at Poincaré Institut in Paris, were later explored by Matta in such paintings as Onyx of Electra and X space and the Ego." 38 Gordon Onslow-Ford : Note sur Matta et la peinture(1937-1941), op. cit., (see note 16), pp. 27-28. "I remember we had been influenced by an exhibition of mathematical objects we had seen at the Trocadero, and also by the Tertium of

	[ Fig-8 ] In Towards a New Subject in Paintings, Gordon Onslow-Ford evoques
	the Palais de la découverte in Paris. « At this time, wrote Gordon
	Onslow-Ford, I remember being impressed by the mathematical objects
	on display at the Palais de la découverte, by the microscopy world,
	photographs of things invisible to the naked eye such as perfumes and
	by a book on morphology of which I have forgotten the title. Also my
	training in physics and mathematics began to take on new meaning 40 »
	Between lines and topographic perspective, the Schwarzschild
	Organum Ouspensky and his theories on humans, one, two, three or four
	dimensions."
	39 Germani Ferrari, op. cit., (see note 2), p. 77. "The unique opportunity to 40 Gordon Onslow-Ford : Towards a New Subject in Paintings, op. cit., (see note 4),
	understand light and time is with a mathematographic memory." and p. 116 : "If p. 12.
	one could imagine going very far, deeper into the mental state, into mental geology, 41 Tree different paintings are titled Children's Fear of Idols 1943-1944.
	we would see an incredible energy bubbling and running forever in each species. 42 Roberto Matta : Composition in Magenta (The End of Everything), 1943, oil on
	Through mathematics, languages, it could happen to detach them from each other, to canvas, 71 x 91 cm, was sold by Sotheby's, New York, on Tuesday, November 15,
	identify them, to name them." 1994.

36

See Christian Zervos : Mathématiques et art abstrait, in : Cahiers d'Art, vol. 11, no. 1-2, 1936, pp. 4-10. The same issue included reproductions of "mathematical objects" photographed by Man Ray at the Institute Poincaré in Paris. See also Jeanine Plottel : The Mathematics of Surrealism, in : The Romanic review, vol. 71, no. 190, n° 3, Columbia University, New York, May 3, 1980. Finally, Max Ernst painted in the U.S., a painting entitled Mattamatichs tribute to Matta in 1942 and reprinted in Patrick Waldberg : Max Ernst au pays qui lui ressemble, in : Cahiers d'Art, year 24, Paris, 1949, p. 290. 37 William Rubin limits his analysis of Matta's paintings to the visit to the Institut Poincaré and unfortunately does not continue the reflection about Schwarzschild singularity. See William S. Rubin : Dada, Surrealism, and Their Heritage, op. cit., (see note 23), p. 160. "singularity expresses a path to the inside space. This mathematical figure is the statistic's shape of what may be a black-hole. In his drawings from 1937-1939 and paintings, Matta used it to suggest the possibility of using doors, passages for some kind of related points, passages for ideas, concepts, through time and space. These métaphorical doors, which are passages through space and time, bring objects together in a new kind of surrealist collage. These spatiotemporal deformations are found in Matta's paintings, when the star is isolated by a dark circular area where color is absent. The paintings titled Children's Fear of Idols 41 , The End of Everything) 42 ,

  Gestes et opinions du docteur Faustroll, pataphysicien, Stock, Paris, 1923. (The Acts and Opinions of Dr. Faustroll, Pataphysician). 56 See the book by Fariba Bogzaran : Matta : Explorer of Hyperspace and his Friendship with Gordon Onslow Ford, in : Matta Onslow Ford : Shared Vision, California, Weinstein Gallery, San Francisco, June 2003. is beautiful enough. […] No fruit, no bottle, no table, no tree. I will not put them on canvas. Never. I swear it." 57

  of the first to consider psychology as a branch of physiology. André Breton and the surrealists are interrested in thoses theories 65 and Breton bought the book of Comte de Tromelin about a response to the book of Haeckel named Les énigmes de l'univers 66 published in 1902. In an other hand, in the International Surrealist Bulletin 67 , published in 1936, Hugh Sykes Davies made a lecture titled Biology and Surrealism where he explained the link between psychoanalyses and biology 68 . Solid structures infinitesimal are dying and they are starting to be an organic system or in an other way, a star explode in a cosmic history. We can say Pompidou, 1985, p. 306. « La peinture a un pied dans l'architecture un pied dans le rêve ». 70 Roberto Matta, Matta [exposition, Paris], Centre Georges Pompidou, Musée national d'art moderne, 3 octobre-16 décembre 1985. Paris: Centre Georges Pompidou, 1985, p. 306. « Il faut présenter les tableaux comme des cartes géographiques de la nature humaine et de ses énergies ».

	development (phylogeny). His main interests lay in life development between architecture and organic systems. Radiolaria found throughout
	processes in general including Darwin's theory and developments of the ocean are amoeboid protozoa that produce intricate mineral
	non-random forms, which is synthesized in the illustrated Kunstformen skeletons, typically with a central capsule which split the cell into inner
	der Natur (Art forms of nature) 62 . Haeckel wrote also his famous and outer portions. In the paintings of Matta, we can often find the
	Generelle Morphologie der Organismen, allgemeine Grundzüge der geometric sphere which explodes itself and in thoses spheres, organites
	organischen Formen-Wissenschaft, mechanisch begründet durch die are crawling and seemed trying to escape till they are liberated out of
	von Charles Darwin reformirte Descendenz-Theorie 63 , published in this capsule. The skeleton or architecture could be a kind of nest in
	1866, presenting a synthesis of Darwin's ideas with the German which life (and also the dreams in a psychic acceptance) is painted in
	tradition of Naturphilosophie going back to Goethe and will be very the steps of it's own transformation till liberate itself from the
	inspiring for the Freudian conception of the individual psyche and the
	collective thought 64 . Haeckel used morphology to reconstruct the
	evolutionary history of life using embryology as evidence of ancestral
	relationships. Haeckel's aim was a reformed morphology with
	evolution as the organizing principle of a cosmic synthesis unifying
	myths, religion, science, and art. In that sense, Ernst Haeckel was also
	62 Haeckel Ernst, Kunstformen der Natur, Hundert Illustrationstafeln mit
	beschreibendem. Text, allgemeine Erläuterung und systematische Übersicht, Leipzig 65 See André Breton, « Réserves quant à la signification historique des investigations und Wien : Verlag des Bibliographisches Institut, 1899-1904 (new edition : München sur le rêve », in Le surréalisme au service de la révolution, n° 4, décember 1931, p. 7-: Prestel, 2012). (English version : Haeckel Ernst, Art forms in nature, New York : 12. Dover Publications, 1974). The published book of Ernst Haeckel includes over 100 detailed, multi-colour illustrations of plants, animals and sea organisms. 66 Le Goarant de Tromelin, Gustave, Comte de, Les Mystères de l'univers, réponse
	aux "Enigmes de l'univers", de Haeckel, Paris, Bibliothèque universelle Beaudelot, 63 Haeckel Ernst, Generelle Morphologie der Organismen : allgemeine Grundzüge 1907. (see André Breton's library sold in 2003) ; and Haeckel Ernst, Les énigmes de der organischen Formen-Wissenschaft, mechanisch begründet durch die von Charles l'univers, traduit de l'allemand par Camille Bos, Paris : Schleicher frères, 1902. Darwin reformirte Descendenz-Theorie, Berlin : G. Reimer, 1866 (French version : 67 International Surrealist Bulletin, n° 4, september 1936, p. 13-18. Histoire de la création des êtres organisés d'après les lois naturelles : conférences scientifiques sur la doctrine de l'évolution en général et celle de Darwin, Goethe et 68 This link was studied in an article of the biologist Jean Vovelle in 2007. See Jean
	Lamarck en particulier; traduites de l'allemand, par Ch. Letourneau; et précédées Vovelle, « Biologie et surréalisme : H. Sykes Davies 1936 », in Mélusine, n° XXVII,
	d'une introduction biographique par Charles Martins, Paris : C. Reinwald, 1874). Lausanne, L'Age d'Homme, 2007, p. 121-134.
	64 Jacqueline Duvernay Bolens, « La théorie de la récapitulation de Haeckel à Freud 69 Roberto Matta, Matta [exposition, Paris], Centre Georges Pompidou, Musée
	», Topique, 2001/2 n° 75, p. 13-34. national d'art moderne, 3 octobre-16 décembre 1985. Paris: Centre Georges

one When Matta developed his Psychological morphologies, it includes the possibility of a biological process that places man in an infinite becoming (ontogeny) and at the same time, in a history of humanity in perpetual becoming (phylogeny). Matta painted shapes that are planets and/or maybe cells or primitive organisms. Matta chose the Radiolaria forms in the 1990's as we can see in the center of several drawings and paintings, because it is a kind of link between architecture (metaphore of ontogeny) and dream or life 69 (metaphore of phylogeny), or so constraints. The sphere is almost a planet or a cell which is in a mutation of its own structure. Matta say it very clearly: « We must present the paintings as geographic maps of human nature and its energies 70 ». In the painting titled Yoniverso (1998) (I am universe) or in Autoportrait des complexités de l'être (1999) Matta shows the structures completly destroyed and the « last » sphere of that universe is going to be deleted to a chaos of forms and colors.

Letter from Matta to Breton, August 15 1942, bibliothèque littéraire Jacques-Doucet, Paris, BRT.C. 1165.

 14 The books are Édouard Monod-Herzen : Science et esthétique. Principes de morphologie générale, Gauthier-Villars, Paris, 1927 and Édouard Monod-Herzen : Art et inconscient, Sonor, Genève, 1932 (a copy was in the Breton's Library, sold in 2003

).15 See letter from Matta to Breton, January 31 1940 (sent from New York to Poitiers), bibliothèque littéraire Jacques-Doucet, Paris, BRT.C.

2097.16 Gordon Onslow-Ford : Note sur Matta et la peinture(1937)(1938)(1939)(1940)(1941), in : Matta, exhib. cat. Paris, Centre Georges Pompidou, Musée national d'art moderne, October 3 -December 16, 1985, p. 32. "The colors are symbolic: blue = sky, yellow = earth, green = vegetation, red and pink = the flesh."

See Gordon Onslow-Ford : Note sur Matta et la peinture (1937-1941), op. cit., (see note 16), p. 35.

Germana Ferrari : op. cit., (see note 2) p. 58. "It was the subconscious in a hot molten liquid [...]. Dreams are images of our volcanic experiences."

LIST OF ILLUSTRATIONS


GEEPS/ECO2/SE Jérôme Juillard

Analysis of resonant pull-in of micro-electromechanical oscillators

In this paper, the equations governing the pull-in of electrostatic MEMS (microelectromechanical systems) oscillators are established and analyzed. This phenomenon defines the maximal oscillation amplitude that can be obtained without incurring instability and, hence, an upper limit to the performance of a given device. The proposed approach makes it possible to accurately predict pull-in behaviour from the purely resonant case, in which the electrostatic bias is very small, to the static case. The method is first exposed in the case of a parallel-plate resonator and the influence of the excitation waveform on the resonant pull-in characteristics is assessed. It is then extended to the more complex case of clamped-clamped and cantilever beams. The results are validated by comparison with transient simulations.

Introduction

The static pull-in of capacitive MEMS structures is a well-studied phenomenon, theoretically and experimentally [START_REF] Senturia | Microsystems Design[END_REF][START_REF] Osterberg | M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures[END_REF]. In particular, it is well-known that a voltage-controlled parallel-plate capacitor may not be displaced by electrostatic forces to a position exceeding one third of the gap without incurring instability. The validity of this result is limited to the case when the bias voltage applied to the capacitor is increased quasi-statically or when the structure is critically damped. Several works have been concerned with establishing similar results to the case when the structure is under-damped, mostly when the voltage is applied as a step [START_REF] Nielsonn | Dynamic Pull-In of Parallel-Plate and Torsional Electrostatic MEMS Actuators[END_REF][START_REF] Krylov | Lyapunov Exponents as a Criterion for the Dynamic Pull-in Instability of Electrostatically Actuated Microstructures[END_REF][START_REF] Leus | On the Dynamic Response of Electrostatic MEMS Switches[END_REF] or when it is a superposition of a DC and of an AC component [START_REF] Alsaleem | An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically[END_REF][START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF][START_REF] Nayfeh | Dynamics of MEMS resonators under superharmonic and subharmonic excitations[END_REF]. The latter case is of great importance for resonant sensing applications, in which the resonance frequency of a structure is measured to track changes in a physical quantity. In this context, it is important to determine the largest stable oscillation amplitude since it determines the maximal magnitude of the detected signal, and, consequently, sets the constraints on the design of the associated electronics of the MEMS/electronics interconnection scheme [START_REF] Kaajakari | Nonlinear limits for single-crystal silicon microresonators[END_REF][START_REF] Arndt | System architecture and circuit design for micro and nanoresonators-based mass sensing arrays[END_REF].

Fig.1 -Open-loop (a) and closed-loop (b) techniques for resonance frequency measurements. In the open-loop

technique, the resonator is typically excited with a chirp. The amplitude and phase response of the system are obtained by demodulating the excitation and the resonator output. In the closed-loop technique, the resonator is excited with the signal output by an electronic circuit enforcing some self-oscillation conditions in the loop. The feedback electronics may include a phase-locked loop in order to guarantee a specific input-output phase relationship, as symbolized by the dotted arrow, or may have a purely feed-forward architecture.

The measurement of the resonance frequency of a MEMS structure can be performed using open-loop or closed-loop techniques (Fig. 1). This paper is dedicated to the latter case, in which feedback electronics sense the position of the resonator and actuate it close to its resonance frequency by enforcing a set of so-called "self-oscillation" conditions (Barkhausen criterion) [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF]. Although the electrostatic closed-loop scheme is ubiquitous [START_REF] Roessig | Surface-micromachined resonant accelerometer[END_REF][START_REF] Nguyen | An integrated CMOS micromechanical resonator high-Q oscillator[END_REF][START_REF] Wojciechowski | A MEMS resonant strain sensor operated in air[END_REF][START_REF] Bahreyni | Oscillator and frequency-shift measurement circuit topologies for micromachined resonant devices[END_REF][START_REF] Verd | Monolithic CMOS MEMS Oscillator Circuit for Sensing in the Attogram Range[END_REF][START_REF] Trusov | Silicon Accelerometer with Differential Frequency Modulation and Continuous Self-Calibration[END_REF], its stability has received little attention so far. Most papers dedicated to resonant pull-in are concerned with open-loop techniques [START_REF] Alsaleem | An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically[END_REF][START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF][START_REF] Nayfeh | Dynamics of MEMS resonators under superharmonic and subharmonic excitations[END_REF], in which the excitation frequency is controlled and the amplitude and phase of the displacement are free to vary. Only a handful of papers [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF][START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF][START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF] set and the amplitude and frequency vary. Of these, [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF] is dedicated to the case when the resonator is a parallel-plate capacitor actuated with a square-wave voltage and is only valid when the DC bias applied to the structure is very small compared to the static pull-in voltage. These three limitations (geometry, waveform and bias voltage) prevent using the results from [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF] in the majority of resonant MEMS/NEMS applications. Such applications are most often based on sine-wave actuation, operate at finite bias voltage and have flexible moving parts, very far from the plane-capacitance hypothesis. In [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF], the framework of [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF] is expanded to account for other nonlinearities (caused by strainstiffening) and the study is made for large DC bias. However, as shown in the present paper, the approach used by the authors of [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF] does not take into account the interdependence of the static component of the displacement and the oscillation amplitude. Consequently, this approach leads to qualitatively and quantitatively inaccurate results, especially at large DC bias. In [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF], resonators of various shapes (parallel-plate, bridge or cantilever beams) with different types of actuation waveforms (square-wave, sine-wave, pulses) and large DC bias are considered. However, the study is limited to the case of symmetric double-sided actuation, in which the static component of the displacement is equal to zero. In an electrostatic MEMS resonator, asymmetry may result from several causes, the most common being that a single electrode is used to actuate the resonator and sense its position; in the present paper, we show that, in such asymmetric cases, the static component of the displacement (resulting from the asymmetry) plays a key role in the pull-in phenomenon.

In this paper, the case of a resonator (parallel-plate, bridge, or cantilever) actuated from only one side is considered and the influence of the excitation waveform on the resonant pull-in amplitude is studied. The main contributions of the paper are that -a general methodology, based on harmonic balancing, for deriving the resonant pull-in condition in asymmetric electrostatic resonators is derived. -this theory is valid from zero DC bias (purely resonant pull-in, as in [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF]) to large DC bias (static, non-resonant pull-in). -the resonant pull-in conditions are given in terms of oscillation amplitude, oscillation frequency and static component of displacement for common resonator geometries and excitation waveforms. -voltage pull-in conditions are also established for these geometries and waveforms, in terms of maximal actuation voltage versus bias voltage. -it is shown that, as in the case of double-sided actuation [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF], the greater travel range for a given DC bias is obtained with pulse-actuation, regardless of the geometry of the resonator.

The outline of the paper is the following. In section 2, the framework and the model governing the transient dynamics of a parallel-plate resonator are described. In section 3, the equations governing the averaged dynamics of the model are established, based on a describing function approach [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF]. Section 4 presents the method for deriving the resonant pull-in characteristics from the averaged dynamics. The analysis is based on an extension of one of the stability criteria presented in [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF] to the case when the displacement of the structure has an amplitude-dependent DC component. This is illustrated in the case of square-wave actuation and show that the method proposed in [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF] for determining resonant pull-in is faulty in the case of a finite DC bias. Section 5 discusses the limits of this work and its extension to more complex cases, in particular to the case of clamped-clamped and cantilever beams. Section 6 contains some concluding remarks.

Framework

We consider the one-sided closed-loop electrostatic actuation of a parallel-plate resonator with stiffness K , mass M , natural pulsation 0 / KM   and quality factor Q . An electrode with surface area e S is situated across a gap 0 G from the resonator, supposed to be very small with respect to the lateral dimensions of the electrode.

A voltage       1 b V t V v t 
is applied across the gap and exerting an electrostatic force on the resonator and setting it into motion. Throughout the paper, the actuation voltage is assumed to be very small with respect to the bias voltage (i.e.

v

). The influence of this hypothesis is illustrated in section 4.3.

The position of the resonator as it moves across the gap is given by     0 G t G a t  , so that 1 a  means that the structure touches the opposite electrode (Fig. 2). , and defining the non-dimensional bias voltage as:

2 0 3 0 2 eb SV KG    , (1)
where 0  is the permittivity of vacuum, the dynamics of the resonator are governed by: 2 1 1

av aa Qa          (2)
If 1 v  , as assumed, the dynamics simplify to:

      2 12 12 1 av a a v N a Q a        , (3)
where   Na represents the dependence of the electrostatic force with respect to a . Throughout the rest of this paper, the tilde is dropped and t is used to describe non-dimensional time.

It is assumed that

  0 sin a A A t  
and that the (non-dimensional) actuation voltage v is in quadrature with a so that an oscillator loop is formed. As we will show in sections 3 and 4, the static component of the displacement, 0 A , plays an important role in the stability of the oscillator. This makes the analysis of single-sided resonators more involved than that of double-sided resonators [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF].

The following three waveforms are considered, which can easily be generated in closed loop when a or a is measured:

-Square-wave:

    0 sign cos v v t   .
-Sine-wave:

  0 cos v v t   . Anchor Flexure K Mobile electrode (M) G 0 G 0 -G(t) G 0 a(t) V(t) (a) (b)
-Pulses:

  0 ,, v v a a   , where   ,, aa  is equal to  
sign a for a duration 2

 

when a crosses the threshold 0 A , and is equal to 0 otherwise. The three waveforms, represented in Fig. 3, can be used to make the resonator oscillate, because the first harmonic of v is, by design, in quadrature with a (the out-of-quadrature case is addressed in section 5). Square and sine waveforms are the most commonly used. Pulsed actuation was introduced in [START_REF] Colinet | Actuation of resonant MEMS using short pulsed forces[END_REF] and studied in [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF][START_REF] Colinet | Actuation of resonant MEMS using short pulsed forces[END_REF][START_REF] Juillard | Large amplitude dynamics of micro-nanomechanical resonators actuated with electrostatic pulses[END_REF]. Pulses can be efficiently generated by high-pass filtering the sign of the detected signal, as described in [START_REF] Brenes | Electrostatically induced modal crosstalk phenomena in resonant MEMS sensors[END_REF]. Among the advantages of pulse-actuation is the possibility to reach very large oscillation amplitude regimes (hence, large signal-to-noise ratios) [START_REF] Juillard | Large amplitude dynamics of micro-nanomechanical resonators actuated with electrostatic pulses[END_REF]. Also, in capacitive MEMS applications, the detected signal is often a superposition of the useful, motional signal (proportional to a ) and of a parasitic signal (proportional to v ) due to capacitive feedthrough. This feedthrough phenomenon makes it difficult to guarantee quadrature between v and a when sinewave and square-wave actuation are used. When pulsed actuation is used, this problem is irrelevant, because the sign of av   (where 0   represents an unknown amount of capacitive feedthrough) is the same as the sign of a , no matter how large v  may be [START_REF] Juillard | Large amplitude dynamics of micro-nanomechanical resonators actuated with electrostatic pulses[END_REF]. This simplifies the design of the oscillator [START_REF] Brenes | Electrostatically induced modal crosstalk phenomena in resonant MEMS sensors[END_REF] and can also be put to use for performing open-loop characterization of the resonator. However, the pulsed-actuation technique is practically limited by the amount of energy that can be supplied to the resonator at each period (the shorter the pulse, the larger 0 v must be), and by the fact that it may become difficult to generate well-defined pulses at very high frequencies (say, greater than a few tens of MHz).

The static pull-in position spi a is defined as the position for which the equilibrium of (3) becomes unstable, when v and all time derivatives equal 0. It is well-known [START_REF] Senturia | Microsystems Design[END_REF] that

1 3 spi a  (4)
and that this position is reached for: 3 Steady-state dynamics 1 Determination of the steady-state regime Describing function analysis [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF] can be applied to (3) to determine the existence of steady-state solutions. We define:

  2 0 2 DC f N a dt       ,     2 sin 0 sin f N a t dt        ,       2 cos 0 0 cos vt f N a t dt v        . ( 6
)
These quantities have analytical expressions that are summarized in table 1, where the following notation is used:

0 1 A R A   .
Although it is not mandatory to perform this change of variables to analyze the system, the use of the "reduced amplitude" R proves to be very convenient, mostly because R varies between 0 and 1, whereas A varies between 0 and 0 1 A  , which is quantity that depends on the bias level  applied to the system.   , as in [22], i.e. that the pulses have a short duration with respect to the period of the oscillation.

DC f sin f     3/ 2 2 2 0 1 11 AR      3/ 2 2 2 0 2 11 R AR  cos f (square wave) cos f (sine wave) cos f (pulses)     2 2 0 4 11 AR       2 1/2 2 2 0 21 1 1 1 AR R          2 0 2 1 A    Table 1 -Expressions of DC f ,

It is remarkable that

  00 ,0 DC B A R f A      , (7-a)       2 sin 0 0 , , 1 0 1 f S A R RA          , (7-b)     cos 00 0 , , 2 0 1 f C A R v R A Q        , (7-c)
corresponding to the averaged dynamics of (3), as derived in Appendix A. This system may have several solutions, the stability of which is studied in section 4. Note that (7-a) defines an implicit relation between 0

A and R (or equivalently between 0 A and A ). The fact that these two quantities are cross-dependent shows that it is not possible to determine the static component of the displacement 0 A through a static analysis of (3) alone.

Practical determination of steady-state response curves

The solutions of ( 7) have no closed-form expressions in terms of the parameters  , 0  and Q , so they must be worked out with a numerical root-finding procedure (e.g. Newton-Raphson). Alternatively, one may proceed as follows:

1. Assume Q is known.

Choose a value of

R between 0 and 1 and  between 0 and spi  .

3. Solve (7-a) for 0 A . Pick the solution (if any) between 0 and spi a .

4. Solve (7-b) for  .

5. Solve (7-c) for 0 v .

The only complex step in the procedure is step (3), which requires solving a polynomial of the third degree (this can be performed numerically or analytically), while steps ( 4) and ( 5) are immediate. If the polynomial of step (3) has no root between 0 and spi a , one should set 0 v to "not a number", meaning that no oscillation is possible for the chosen values of Q , R and  . After repeating this process for different values of R and  , one may then plot the color maps or contour lines of

  0 , vR  ,   0 , v  or   00 , vA  .
As we have just shown, solving for the steady-state is rather complex. However, the resonant pull-in condition can be established with more ease, as illustrated in the following sub-section.

Determination of the resonant pull-in condition 1 Stability of the steady-state regime

Let us denote by   * * * 0 ,, AR a solution of [START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF]. A necessary and sufficient condition for this equilibrium to be stable is that the Jacobian of (A-14):

1 2 00 S DS Q DR C DC DR R                  J (8)
verifies the Routh-Hurwitz criterion, where the derivative operator

0 0 A D DR R A R       , (9)
accounts for the cross-dependence of 0 A and R . In [START_REF] Kaajakari | Nonlinear limits for single-crystal silicon microresonators[END_REF],

0 0 A BB RA R      (10)
is obtained by applying the implicit function theorem to (7-a). The Routh-Hurwitz criterion consists in a set of inequalities which must be verified by the coefficients of the characteristic polynomial of J ,     3 det P   JI , for the system to be stable:

1 0 C Q     , 1 20 S DS C R DR Q        , 0 DC S DS C DR DR     . ( 11
)
One may verify that, for all waveforms, the first inequality is always met provided 0 Q  . Furthermore, the second inequality is also met provided * This condition can be used to complement the numerical procedure described in section 3-2 and discriminate stable solutions of (7) from unstable ones. However, there is not much physical insight gained from numerically evaluating this quantity, so it is interesting to go one step further and try to derive a resonant pull-in condition from [START_REF] Roessig | Surface-micromachined resonant accelerometer[END_REF].

Resonant pull-in condition

By definition, the system is at the resonant pull-in condition when in a state   0 ,, rpi rpi rpi AR which verifies [START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF] and

0 DS C DC S DR DR     . ( 13
)
The resonant pull-in condition is defined by a set of four equations, ( 7) and ( 13), with three parameters (  , Q and 0 v ) and three unknowns (

0 rpi A , rpi R and rpi  ).
It is then a tedious but straightforward process of algebraic manipulation to remove four of these quantities and obtain a single relation between the two remaining ones. The simplest way to do that, which also leads to the most meaningful relations, is to eliminate  , Q , 0 v and rpi  from ( 7) and ( 13) and derive a waveform-dependent relation of the form

  0 rpi rpi rpi R f A 
. This relation defines an "upper bound" for the solutions of ( 7), in the sense that  

* * * 0 ,, AR is stable provided   ** 0 rpi R f A 
(and unstable otherwise).

In the case of square-wave actuation, it is found that:

rpi rpi rpi rpi rpi rpi rpi rpi RA A A A A A A         , ( 14-b)
and, in the case of pulse-actuation 2 :

0 0 13 16 rpi rpi rpi A R A    . (14-c)
Note that these relations not only predict the occurrence of resonant pull-in but also that static pull-in takes place when

0 1/ 3 rpi spi
Aa  , since in that case we find 0 rpi R  , independently of the waveform. On the other hand, the maximal oscillation amplitude is waveform-dependent: it is obtained when 0 0 rpi A  (a situation which may happen when a very small electrostatic bias is used, as in [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF]). In that case, we find for square-wave actuation:

1 This inequality is derived in [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF] using a more "physical" approach, in the case when 0 0 A  . An alternative approach to the determination of this inequality is given in appendix A. which is a well-known result [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF]. For sine-wave actuation, the maximal oscillation amplitude is found to be: Finally, we find, for pulse-actuation:

1 rpi R  , (15-c)
i.e. that a full-gap travel range is possible if the electrostatic bias is very small. It is interesting to note that these values are the same as those obtained in the case of double-sided actuation [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF].

One can calculate rpi  by combining (7-a) and (7-b) which yields:

* * 0 * 0 13 1 A A     , ( 16-a)
or equivalently

*2 * 0 *2 1 3 A      , (16-b)
which is valid for all waveforms.

It is interesting to note that ( 14) and (7-a) imply that 0 rpi A and rpi R (and, consequently, rpi  ) depend only on 

and not on Q or 0 v . There are no closed-form expressions for  

0 rpi A  ,   rpi R  or   rpi  
, but these quantities are plotted in Fig. 4 for the three considered waveforms. Equation (7-c) can then be used to determine, for a given quality factor and a given waveform, the maximal value of 0 v that can be used without incurring resonant pull-in (i.e. a "voltage pull-in condition", as defined in [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF]), as plotted in Fig. 5. These results are compared to those from [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF] in the following section.

Application to square-wave actuation

In this sub-section, we focus on the case of square wave-actuation. Equation (7-c) can then be rewritten as:

    3 2 00 1 11 8 rpi rpi rpi rpi v A R R Q      . (17)
Equation ( 17) should be compared to the resonant pull-in condition found in [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF], which can be written, using our notations:

  3 00 1 1 12 3 rpi rpi vA Q    , (18)
where, for a given DC bias  , the authors propose to estimate 0 rpi A in [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF] thanks to a static analysis of (1) and to use the following approximation for rpi  :

  3 0 2 1 1 rpi rpi A     , (19)
based on a first-order linearization of (1) close to the static solution.

Transient simulations of (1) are performed with Matlab/Simulink, for different values of  and 10 Q  . For each value of  , the value of 0 v is increased step by step until resonant pull-in is reached. The smallest value of 0 v for which the system is unstable is noted down. These results are compared with those obtained with ( 17) and ( 18) (Fig. 6). As could be expected, ( 17) and ( 18) yield similar results when the DC bias is small ( 1   ) but overestimate the value of 0 v . For finite values of  , the two methods no longer agree: while transient simulation of [START_REF] Senturia | Microsystems Design[END_REF] show that (18) fails to predict the correct resonant pull-in condition for finite values of  , the results obtained with (17) are very accurate. This discrepancy for finite values of  can be attributed to an error made in [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF] in deriving [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF]: it is assumed in [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF] that equation (29), reproduced hereafter with our notations:

  2 3 0 0 8 10 Qv A A A       ( 20
)
is a polynomial of the third degree in A and that resonant pull-in is reached when its discriminant is zero. However, as explained in section 3, 0 A and A are cross-dependent so that (i) one may not derive values of 0 A and  from a static analysis alone and (ii) one cannot consider (20) as a third- degree polynomial.

In the case of small  , the difference between the results obtained with both methods and transient simulations can be explained by the fact that the predicted values of 0 v are then finite and, consequently, fall outside the scope of both studies. However, in the square-wave case, 22 0 vv  is a constant, and the analysis made in the previous sections still holds, provided one multiplies 1 v  in [START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF]. Equations (14-a) and ( 16) remain valid in spite of this modification. Thus, in order to find the maximal value of 0 v that can be used without incurring resonant pull-in, one should (i) choose a value of rpi R between 0 and 1 / 3 , (ii) deduce the corresponding 0 rpi A and rpi  with (14- a) and (16-a), (iii) eliminate 0 v from (7-a) and (7-c) and solve the resulting second-degree polynomial for  , (iv) calculate 0 v from (7-c). The results obtained with this approach are also represented in Fig. 6 and show an excellent agreement with transient simulation results.

Extension to more complex cases and limits of the study 1 Extension to other resonator geometries

Electrostatically-actuated cantilever beam and clamped-clamped beam resonators with rectangular cross-sections are commonplace in MEMS and NEMS applications. Calculating the resonant pull-in condition for these geometries is more arduous than for parallel-plate resonators. Still this can be achieved by numerically approximating the projection of the electrostatic force on the shape function used to describe the beam deformation in a proper way, and, consequently, the functions

    3 4 2 2 0 , 2 42 0 12 2 LL b Ebh w w w V bh x t t Gw                , (21)
where it is supposed that the actuation electrode extends from



. Note also that the electrode/beam length is supposedly large with respect to the gap and to the width of the beam, so that the plane capacitor approximation remains valid in a cross-section of the beam. Setting

x

                . (22)
Letting

      1 , w x t a t w x 
, where   1 wx is the first beam eigenmode, normalized so that

  1 max 1 w 
, a Galerkin procedure is applied to [START_REF] Juillard | Large amplitude dynamics of micro-nanomechanical resonators actuated with electrostatic pulses[END_REF]. This leads to the projected non-dimensional dynamics:

        1 2 1 12 1 wx a a a v dx Q aw x          , ( 23
)
where Approximate expressions for the right-hand side of ( 23) can be obtained as explained in appendix B, guaranteeing that the relative error between the projected force and its approximation is bounded for 1 a  . For example, in the trivial case when 0   and 1   , the projected dynamics become:

42 0 33 10 1 b LV E h G     , 2 2 2 E bh Q L    (24)
    3/2 1 0.017 0.523 1 2 1 aa a a v Q a        , (25)
in the case of a clamped-clamped beam, and

      1 0 783 0.392 1 2 0 531 0 114 log 1 1 a + . a a a v + . + . a -a Qa          (26)
for a cantilever beam. The resulting reduced-order models can then be averaged over one period to yield the equivalent of system [START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF] for a deformable beam. The expressions of DC f , sin f and cos f can be derived as shown in appendix C of this paper (the calculation of DC f is illustrated in appendix D). The same process as in section 4 can then be applied to determine resonant pull-in conditions: these are represented in Fig. 7 and Fig. 8, in the case when 0   and 1   . Note that this last step must be performed numerically: while it is simple to use ( 13) and the steady-state equations to obtain an implicit relation between 0 rpi A and rpi R where only these two quantities appear, it is (to the author's knowledge) impossible to make it explicit, as opposed to the parallel-plate case. Finally, in the more general case when the lengths of the beam and the actuating electrode are different, one should replace   Na in ( 6) by a parameterized reduced-order model   ˆ, Naθ , established as in appendix B, and follow the same procedure as above.

Extension and limits of the study

This study is admittedly limited to an idealized framework and the presented results should always be quantitatively questioned when working in an experimental context. In this section, the main limitations of our study are listed, in order of ascending difficulty, and some guidelines for tackling them are given.

a) Non-ideality of the geometry

First of all, it must be stressed that the reduced-order models proposed in section 5.1 are valid as long as the plane capacitor approximation holds in a cross-section of the beam. However, this hypothesis is not always verified, in particular in nanoscale devices. Geometries with small length/gap or width/gap ratios require either that fringing fields be accounted for in the expression of the electrostatic force (this may also be done with the approach proposed in [START_REF] Juillard | Modeling of micromachined beams subject to nonlinear restoring or damping forces[END_REF]), or that other reduced-order modeling strategies be used, such as the fully numerical approach proposed in [START_REF] Gabbay | Computer-aided generation of nonlinear reduced-order dynamic macromodels[END_REF]. Regardless of which approach is used, the resonant pull-in condition may still be derived from [START_REF] Nguyen | An integrated CMOS micromechanical resonator high-Q oscillator[END_REF], provided the dynamics of the device can be captured with only one mode. In spite of the added complexity of deriving a model of the electrostatic force, no new parameter is introduced in the equations and the resonant pullin condition may still be obtained at no great cost.

Other interesting "non-ideal" geometries are asymmetric double-sided resonators, to which the methods presented in this paper can readily be applied.

b) Presence of other nonlinearities

When other nonlinearities (e.g. caused by beam elongation, squeezed-film damping, etc.) are present, the set of four equations defining the resonant pull-in condition contains more than four parameters. Consequently, it is not possible, in such a case, to derive "nice" closed-form expressions of the resonant pull-in condition, such as ( 14). Yet, it is still possible to use [START_REF] Nguyen | An integrated CMOS micromechanical resonator high-Q oscillator[END_REF] to obtain resonant-pull-in curves parameterized by the supplementary parameters.

For example, beam elongation is equivalent to Duffing stiffening, with a geometry-dependent Duffing coefficient  . If this phenomenon is taken into account, the resonant pull-in condition, can be worked out as previously, by eliminating  , Q , 0 v and rpi  from ( 13) and [START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF]. The resulting relation between 0 rpi A , rpi R is then parameterized by  (the case 0   corresponding to Fig. 7 and8). In the present paper, it is assumed that the displacement of the structure ( a ) or its velocity ( a ) is directly measured and the waveform used for actuation is perfectly in quadrature with a . This makes it possible to "nicely" separate the influence of the actuation voltage from that of the bias voltage in the steady-state equations, and yields closed-form expressions of the resonant pull-in condition 3 .

However a phase delay  may be present in the feedback loop, which also introduces a supplementary parameter in ( 7) and ( 13) and should be treated as in 5.2.b.

For example, detection nonlinearities may make it practically difficult to generate the actuation waveform in quadrature with the displacement. A typical example is when a capacitive detection scheme is used: first of all, the detected signal is polluted by "feedthrough", i.e. the detected signal is the superposition of the motional signal a and of the actuation waveform, which impacts the loop delay. Also, the detected signal becomes heavily distorted as a increases (the capacitance being a nonlinear function of a ), which may make it difficult to ensure quadrature in the loop, unless some particular electronics are used (e.g. a PLL).

Thus, one should use the present study "as is" only when the feedback loop correctly enforces quadrature over the whole range of possible values of a . Otherwise, one should adapt the theory to the non-ideal framework.

d) Interaction with other modes

It should be pointed out that the present study is limited to the case when the resonator can be modeled as a system with a single degree-of-freedom. It should be used with caution for complex structures because of possible modal interactions induced by electrostatic coupling. In particular, as a increases and becomes finite, it may be incorrect to neglect the components of the electrostatic force at 2 , 3 , etc. and their effect on the dynamics of the system [START_REF] Brenes | Electrostatically induced modal crosstalk phenomena in resonant MEMS sensors[END_REF].

Conclusion

In this paper, a method for determining the steady-state dynamics and the resonant pull-in condition of MEMS-based oscillators actuated from only one side was proposed. After illustrating this method in the case of a parallel-plate resonator, this approach was extended to clamped-clamped beam and cantilever beam resonators. The proposed approach is original as it accurately captures the crossdependence of the static and dynamic components of the resonator displacement. This is in contrast with the double-sided case (in which the displacement has no static component) [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF].

The proposed approach also predicts pull-in for 0 spi   , i.e. from "purely resonant" to static pullin, whereas previous studies of closed-loop single-sided actuation are either limited to the case when the bias voltage is very small with respect to the static pull-in voltage (i.e.

spi  ) and square-wave actuation [START_REF] Seeger | Dynamics and control of parallel-plate actuators beyond the electrostatic instability[END_REF] or fail to accurately predict pull-in [START_REF] Fargas-Marques | Resonant Pull-In Condition in Parallel-Plate Electrostatic Actuators[END_REF], as comparison with the proposed method and transient simulations has shown. The increased displacement range obtained when using pulsed actuation compared to sine-wave or square-wave actuation is also an interesting outcome of this study.

In the case of flexible beam resonators, the proposed approach is used in conjunction with a singlemode approximation of the nonlinear actuation forces, which is valid across the whole gap [START_REF] Juillard | Modeling of micromachined beams subject to nonlinear restoring or damping forces[END_REF], and resonant pull-in conditions are also established. It should be noted that the calculations resulting from this approach are simpler than those which would be obtained with the reduced-order modeling approach proposed in [START_REF] Alsaleem | An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically[END_REF], which requires several modes to capture the large-amplitude behavior of electrostatically-actuated MEMS beams 4 . Moreover, the reduced-order modeling approach proposed in [START_REF] Alsaleem | An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically[END_REF] is inaccurate when used with only one mode (dramatically so if used in conjunction with oneharmonic harmonic balance analysis, as shown in [START_REF] Juillard | A comparative study of reduced-order modeling techniques for nonlinear MEMS beams[END_REF]). Because of this added complexity, and also because the open-loop problem is more involved than the closed-loop problem, no resonant pull-in conditions are established in [START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF][START_REF] Nayfeh | Dynamics of MEMS resonators under superharmonic and subharmonic excitations[END_REF].

Consequently, extending the method of the present paper to the open-loop case, in which the frequency is set and the phase varies, is the subject of ongoing work. Future work will also be devoted to accounting for non-idealities described in section 5-2, in particular to feedback loop non-idealities such as phase-delay and capacitive feedthrough.

Appendix A -Averaged dynamics of the system

Let       0 sin a t A A t t   , (A-1)
where   At and   t  are assumed to be slowly-varying functions of time. Following [START_REF] Gelb | Multiple-input describing functions and nonlinear system design[END_REF], we define

A A   , (A-2)
the rate of change of the amplitude and

  . (A-3)
Differentiating (A-1) yields:

  sin cos aA      , (A-4)       22 sin 2 cos aA             . (A-5)
Injecting (A-1), (A-4) and (A-5) in (3) and taking the average, the projection on sin and the projection on cos over one period of the resulting equations leads to:

0 DC Af   , (A-6)   22 sin 1 f AQ           , (A-7) cos 0 22 f v AQ        . (A-8)
Equations (A-2), (A-7) and (A-8) are a nonlinear autonomous dynamical system. The equilibrium is reached when the time-derivatives equal zero, i.e. in all of these equations, since

    0 0 1 1 AR AR A A R R       . (A-13)
Using this notation, the dynamical system can be written, with the notations defined in section 3, as:

    2 0 0 ,, , , 2
S A R Q C A R RR                     . (A-14)
The stability of the system is studied in section 5, by applying the Routh-Hurwitz criterion to the jacobian matrix of (A-14).

Alternatively, one may derive a stability condition by assuming that a is given by:

  0 cos 1 cos a A V A     (A-15)
instead of (A-3), where V would be a reduced velocity. The dynamics of the system are then governed by:

  cos 00 0 2 , , 1 f V V v A R V AQ      , (A-16)     2 sin 0 2 0 , , 1 0 1 f V A R V R A R          , (A-17)
while (A-6) remains unchanged. The stability of this dynamic system is guaranteed provided

0 0 0 A R V V R V A V                . (A-18)
Taking the total derivative of   0 ,,

A R V  with respect to V yields: R D V DR V       , (A-19)
where the differential operator D is the same as in section 4: Furthermore, we have:

  00 , , , , V A R V S A R R     . (A-25)
Finally, the jacobian determinant of the transformation from   , R  coordinates to   , RV coordinates is positive. Thus, (A-22) is equivalent to [START_REF] Roessig | Surface-micromachined resonant accelerometer[END_REF].

Appendix B -Reduced-order modelling of MEMS beams

By and large, the reduced order models given in the paper are obtained as in [START_REF] Juillard | Modeling of micromachined beams subject to nonlinear restoring or damping forces[END_REF], by minimizing the relative error between a numerical estimation of the projection of the electrostatic force on the first beam eigenmode, denoted by

1 w :         1 2 1 1 wx dx aw N x a      (B-1)
and a model parameterized by θ ,   ˆ, Naθ . The relative error is summed over a set of chosen modal amplitudes i a , so that the optimal parameters are given by:

    2 ˆ, arg min 1 i opt i i Na Na       θ θ . (B-2)
For a clamped-clamped beam, provided 1

 

(the actuation electrode is centered), the model has the following form:

      3/2 1 1 ˆ, CC N aa a N    . (B-3)
For a cantilever beam, provided 1   (the actuation electrode extends to the tip of the beam), it can be chosen as:

      1 1 2 3 2 3 1 1 ˆ, , , log 1 CA a N a a a N a                  . (B-4)
The coefficients

 

Na, as shown in [START_REF] Juillard | Modeling of micromachined beams subject to nonlinear restoring or damping forces[END_REF], so that the relative error is bounded across the whole gap.

To obtain the parameters appearing in ( 25) and ( 26), (B-2) is solved: this is a trivial linear least squares problem. The error is minimized over a set of 100 values i a of the modal amplitude, uniformly spaced between -0.99 and 0.99.

Introducing the log function in the ROM of the cantilever beam (B-4) gives better results than the approach proposed in [START_REF] Juillard | Modeling of micromachined beams subject to nonlinear restoring or damping forces[END_REF]. This approach was already used in [START_REF] Juillard | Influence of excitation waveform and oscillator geometry on the resonant pull-in of capacitive MEMS oscillators[END_REF].

Note that the exact values of the model parameters depend not only on the chosen abscissas, but also on the geometry of the actuation electrode. The numerical values given in the paper are only valid when the actuation electrode is the same length as the beam. However, the approach of [START_REF] Juillard | Modeling of micromachined beams subject to nonlinear restoring or damping forces[END_REF] can also be used to approximate integrals of the form: In particular, the following equality is useful for determining

    1
DC f :     2 2 21 0 1 1 , ,1, 2 22 1 sin n dt nn F R Rt          , (C-1)
which is valid for any real n .

We also have:

      2 2 0 1 1 log log 1 sin 11 2 2 dt Rt R          , (C-2) and           2 2 2 0 1 1 log 1 1 log 1 sin 1 sin 11 2 2 dt R R t R t R             . (C-3)
sin f can be determined in the same fashion using:

      2 2 21 0 sin 1 12 , , 2, 22 1 sin n t dt nn nR F R Rt          , (C-4)       2 2 0 12 sin log 1 sin 11 R t dt Rt R       , (C-5) and             2 2 2 0 1 1 1 1 sin log log 1 sin 1 sin 11 2 2 11 t dt R R t R t R R                 . (C-6)
It is immediate to derive the waveform-dependent function cos f in the case of pulse actuation. For the square-wave case, the following formulas must be used:

                2 11 0 cos 1 2 1 1 cos 1 11 1 sin n n n t dt sign t nR RR Rt             , 1 n  , (C-7)           2 0 cos 1 2 1 cos log 1 1 sin t dt R sign t RR Rt           , (C-8)                     2 0 12 cos cos log 1 log 1 1 log 1 2 1 sin t sign t dt R R R R R Rt R            , (C-9)                         2 22 0 11 cos cos log 1 log 1 1 log 1 2 1 sin 1 sin t sign t dt R R R R R R t R t R            . (C-10)
Finally, in the case of sine-wave actuation, we have: Note that (C-1), (C-4) and (C-11) can be simplified when n is an integer, as in the parallel-plate case and in the cantilever case, or a half-integer, as in the clamped-clamped case. The corresponding expressions, tabulated from [24], are given in Table C Provided the actuation voltage v is in quadrature with the displacement, taking the average of (D-4) over one period of  yields, using (C-1) in the case where K (resp. E ) stands for the complete elliptic integral of the first (resp. second) kind, with argument

      2
      2 81 K 1 E 1 R R R        2 8 1 E 1 K 3 RR R     1 n  2 1 1 R    22 2 1 1 1 RR    2 2 11 R  3 2 n  2 21 E 1 R R        22 81 E 1 K 31 R R RR         2 81 K 1 E 1 R R R   

 

2 / 1 RR  , as in table C-1 .

Similarly, for a cantilever beam, the non-dimensional electrostatic force projected on the first mode may be expressed as: The expressions of sin f and cos f may be derived in a similar fashion.

     

  are dedicated to resonant pull-in in closed-loop, in which the phase is

Fig. 2 -

 2 Fig. 2 -Parallel-plate resonator with no voltage applied (a) and in a deflected position (b).Letting a dot (as in a ) denote differentiation with respect to

Fig. 3 -

 3 Fig. 3 -From left to right: square (a), sinusoidal (b) and pulse (c) waveforms (continuous line), for 0 1 v  . The dashed line represents

  three waveforms of Fig.3. The expression of cos f in the case of pulse-actuation uses the assumption that 1

  of pulse actuation. The steady-state regime is determined by solving the nonlinear system of equations:

  0 spi Aa  . Since these two conditions are always met in practice, the stability of the oscillation boils down to:

2

  In that case, we find that 0CC   , and the pull-in condition[START_REF] Nguyen | An integrated CMOS micromechanical resonator high-Q oscillator[END_REF] simplifies to:

FigFig. 5 -0

 5 Fig. 4 -

Fig. 6 -

 6 Fig. 6 -Voltage pull-in condition obtained with three different methods compared to transient simulation results. The dashed curve corresponds to the dotted curve of Fig. 5, translated by one decade (since 10 Q  ).

  example, we start with the Euler-Bernoulli equation, governing the displacement w of an electrostatically-actuated rectangular beam with length L , thickness h , width b , Young's modulus E , viscous damping coefficient  and density  :

Fig. 7

 7 Fig. 7 -

Fig. 8 -

 8 Fig. 8 -Voltage pull-in condition for a clamped-clamped beam (a) and for a cantilever beam (b).

  -ideality of the feedback loop

  e. the model and the projection of the electrostatic force coincide when 0 a  . The structure of the model, independently of its parameters, guarantees that the asymptotic behavior (as a goes to 1) of   ˆ, Naθ is the same as that of

f

  for fringing field effects ( 1 np  ) or other nonlinear forces. in the case of cantilever or clamped-clamped beams, Naθ obtained in appendix B. The resulting integrals can all be expressed in terms of Gauss hypergeometric function 21 F [24].

  As an example, consider a reduced-order model for a clamped-clamped beam with a centered electrode of arbitrary length: -1) holds, the right-hand side of (D-2) becomes:

1 ) 7 )

 17 holds and making use of the reduced amplitude, (D-6) becomes:Provided the actuation voltage v is in quadrature with the displacement, taking the average of (D-7) over one period of  yields, using (C-1) in the case 1 n  , (C-2) and (C-3):

  -1.

		21 F	2  1 , ,1, 22 nn R   	21 F	2  12 , , 2, 22 nn R   	2  1 , , 2, 22 nn FR 21   
	n 	1 2	21 K 1 R

Table C -

 C 1 -Expressions of Gauss' hypergeometric function for half-integer values of n (clamped-clamped beams) and for1 n  (cantilever beams). In this table, K (resp. E ) stands for the complete elliptic integral of the first (resp. second) kind, with argument

At least for parallel-plate resonators. In the other geometries, the pull-in condition can easily be established using a numerical procedure, as shown in section 5.1 and 5.2.a.

As stated by the authors of[START_REF] Alsaleem | An Experimental and Theoretical Investigation of Dynamic Pull-In in MEMS Resonators Actuated Electrostatically[END_REF][START_REF] Nayfeh | Dynamic pull-in phenomenon in MEMS resonators[END_REF] themselves: "A minimum of three modes were employed in the reduced-order model used to predict the transient and steady-state responses reported here. This corresponds to a sixdimensional phase space. As a result, generating the device stability maps is a nontrivial task."