Literature between Medicine and Religion: Herder's aesthetics of touch and the emerging field of Medical Humanities

This article uses a reading of Herder"s early essay Sculpture to locate Herder"s place within the complex genealogies of thought regarding the mind-body problem. It analyses how Herder discusses sculpture and touch by combining an idealist with a materialist position. Herder theorizes sculpture in order to close the gap between mind and body as well as that between art and life. According to Danto, this non-dualist approach shapes much of the contemporary art scene.

exciting eighteenth century thinker to the reader, the following section will trace the young Herder"s intellectual itinerary from medicine to theology and philosophy.

Herder was unorthodox because he denied the divine authorship of the Bible, believed that revelation was not a prerequisite of the Christian faith but was rather scattered throughout all religions. He moreover did not propound a systematic theology which advanced a Christology or offered a dogmatic conception of sin. As

Clark has succinctly put it, "the simple truth is that Herder"s religion at this time [i.e. ca. 1764] lacked two important doctrines common to most Christian creeds: (1) a systematic doctrine of the nature of sin, and (2) the doctrine of the vicarious sacrifice of Jesus Christ" (Clark,p. 211). Clark goes on to make clear that this state of affairs is by no means restricted to the year 1764: "although Herder was at no time an atheist or agnostic, he was likewise at no time after 1764 a believer in the majority of the doctrines of his church or any other" (Clark,p. 211). This unorthodoxy would certainly have found strong support in the Socinianism which informs Spinoza"s writings on Jesus" political and social impact rather than on the doctrinal core of Christian instruction-be that original sin or redemption through crucifixion (see Clark,p. 211). Like Spinoza, Herder focuses on Jesus" humanity, on his justitia and on his caritas. i Indeed when Herder gave Goethe a copy of Spinoza"s Ethics he draws a parallel between Spinoza"s and Jesus"s respective teaching (see Bell,p. 98). Herder"s theological work was scandalous, because it was radically non-dogmatic. The theologian Friedrich Wilhelm Kantzenbach has rightly emphasized Herder"s heterodoxy. According to Kantzenbach Herder promotes the personal religion of Jesus which is quite different from the organized religion of Jesus Christ the redeemer (Kantzenbach,p. 104). This is not to deny that Herder was a theologian and part of the Lutheran clergy. His understanding of humanity and its history diverges, however, from Christian dogmatic teaching. In order to better understand Herder"s theological position, it worth referring to David Jasper"s distinction between Christendom and the heterodox or radical theology of the early Christianity of the desert fathers: "Only with the radical dissolution of Christendom and its aftermath in the West can the radical theology that has always been the theology of the desert begin to be articulated again in a language whose end is always silent" (Jasper,p. 162). Herder"s theology finds its end in being silent and this nowhere more so than in his work as literary and art critic. The silent or non-assertive presence of religion within Herder"s writing on art and literature is part of its potential appeal to contemporary theologians like Jasper who are sensitive to the presence of religion in what is apparently non-religious. This article analyzes how Spinoza"s medical conception of the mind as the idea of the body informs Herder"s silent theology of embodied works of art (like sculpture). Spinoza is important because he relates the cerebral or spiritual (the mind) to the corporeal. Spinoza"s view establishes an isomorphism between the two entities while at the same time not questioning their distinct identities. As the idea of the body, the mind is responsible for actions performed within the external and material world of embodiment. The mind does, however, also depend on the medicine-based well being of the body. According to Spinoza there is therefore not a hierarchical relation in which one commands the other. Rather the mind has to take care of the body and the body has to take care of the mind in order to ensure the preservation of the self (conatus). Spinoza did not only align the life of the mind with that of the body. He also established an invariable link between the equilibrium of the individual and that of the society to which he or she belongs. As Genevieve Lloyd has put it: "The mind is the idea of a body which is what it is, and does what it does, by virtue of being part of wider wholes reaching up to the totality of the material world" (Lloyd,p. 96). The connection between the biological and the epistemological on the individual scale thus prepares the ground for the larger sphere of inter-subjective relations that connect the preservation of the self to the survival of the other.

In a highly idiosyncratic way Herder revised and, to an orthodox reader undermined, modern Christian thought along Spinozist lines. This non-dogmatic approach also informs the relationship between philosophy, medicine and theology in Herder"s thought. As Clark has pointed out, Herder"s theology is quite liberal compared to Kant"s theological focus on original sin and the old Adam: "Herder, the protestant theologian and church official, defends in his Ideas the idea that man is fundamentally good, spoiled only by over-refinement and specialization (which is a function of reason), while Kant, the liberal philosopher, defends the Protestant, Christian, orthodox view that man because of his lower impulses, is fundamentally evil and has to be saved by an outside agency from the misdirection of his powers" (Clark,p. 320).May there be a connection between Herder"s non-dogmatic theology and his critique of his former mentor, the idealist Kant? In his important book Kant, Herder and the Birth of Anthropology John H. Zammito has shown how Herder was in many ways much more open-minded than the critical Kant. Could

Herder"s Spinozism have caused his intellectual clash with Kant? David Bell, for one, has argued that Herder"s critique of Kant derives from his Spinozism: "Herder"s sense of divine immanence and his naturalistic monism are diametrically opposed to Kant"s rigid dualism" (Bell,p. 143). Herder conceives of God and the world along Spinozist lines and in doing so he delineates the outlines of what I will later call an immanent alterity. Here we find the extraordinary in the ordinary (as in the embodied appreciation of our surroundings via sculpture). This Spinozist approach, however, was certain to offend Christian orthodoxy.

Why did Herder risk causing offense? He was after all trained as a theologian and was thus aware of the scandalous nature of his theological arguments. So why did Herder consciously and conscientiously offend the religious establishment of his time? This question brings us to the reasons that motivated him to study theology in the first place. In contrast to other eighteenth century German thinkers with a protestant background -Friedrich Hölderlin is a famous example of an adolescent intellectual who was obliged to study theology with a view of earning a living as a pastor, (which he finally refused to become)-theological studies were not forced upon the young Herder.

In fact Herder began his University studies by reading medicine. It was his discomfort with his medical training that opened up his interest in theology. What precisely made Herder turn away from his initial scholarly pursuit? Medicine has a strong practical component and it might well have been this balance between practice and theory that attracted Herder in the first place. What alienated him from the desired practical pursuits of a medical doctor was the confrontation with death. In 1762, during the dissection undertaken by a Russian field doctor at the University of Königsberg, Herder passed out in horror. As Simon Richter has put it:

The sight of the dissected body was a painful Reiz for Herder; spontaneous contraction (Zurückziehung) occurred in the form of passing out. He became all body as his consciousness fled. Returning to consciousness, his repulsion for the body decided him once and for all for the study of theology and philosophy. Days later he began attending the lectures and seminars of Immanuel Kant (Richter,p. 106).

The dissection of the body lays bare its veiled but inherent decomposition. Becoming body uncannily instantiates a near death experience: the loss of consciousness seems to introduce the reign of corporeal life, which turns out to be death. On this view, Herder appears to subscribe to Descartes" mind-body divide. Rather than being the idea of the body, the mind escapes from the body"s material entrapment in disease and decay: "What Herder is afraid of is the unavoidable presence of death in the material body. Once again confronted with the body, Herder passes out, and parts company altogether from the body, even as his own body slumps to the ground" (Richter,p. 127). Richter thus offers an intriguing account of Herder"s confrontation with the medical body. This is, however, only one part of his educational itinerary. True, confronted with the decayed body, as so forcefully presented in the dissection theatre, Herder abandoned his medical training and took up the study of theology and philosophy under Kant"s generous mentorship. What attracted him immensely to the pre-critical Kant was independence of thought as well as the all-inclusive quest for truth. Even after having taken issue with Kantian transcendental philosophy, the mature Herder of the Letters toward the Advancement of Humanity praised Kant as an inspiring teacher: "no intrigue, no sect, no advantage, no nomenclatural ambition (Namen-Ehrgeiz) ever held out for him the least attraction (Reiz) compared to the expansion and enlightenment of truth. He encouraged and pleasantly extorted independent thinking (Selbstdenken); despotism was not part of his personality (Gemüt)" (Herder, 1991, p. 424). When Herder took issue with what he perceived to be the dogmatism of Kant"s transcendental philosophy, he might well have believed himself to act according to the teachings of his former mentor. His critique of the post-1769 Kant thus enacted the instructions of the pre-1769 Kant: namely, independent thinking and the nonexclusive quest for enlightenment.

Yet this enthusiastic search for truth opens up an abyss of lifelessness which in a perturbing way resembles Herder"s traumatic experience in the dissection theatre.

In 1764 Herder finished his theological and philosophical studies in Königsberg and left for the Baltic city Riga where he worked as a writer, priest and teacher. From this far off and rather marginal place Herder established himself as an outstanding German literary critic. Herder, the son of a lower class family, became a prominent public figure by dint of his work as a writer. Neither his lowly origins nor the marginality of the place in which he wrote dampens the appeal of his literary output. Here he is clearly the pupil of Kant: diffidence about his economic and class background does not deter him from gaining recognition of his intellectual achievements.

Why does Herder nevertheless feel uncomfortable as a writer? He enters a space that seems to be free of domination. Jürgen Habermas has famously called this space the "public sphere". Here different arguments compete for prominence on purely rational grounds, regardless of the social and economic status of its participants: "The "domination" of the public, according to its own idea, was an order in which domination itself was dissolved; veritas non auctoritas facit legem" (Habermas, p. 82). On this view, the emerging eighteenth century public sphere subjects domination to the powers of reason. This explains why Herder could achieve public prominence through his Fragments about New German Literature. It does, however, not account for the widespread appeal of his work as a literary critic.

The fact that Herder developed his popular philosophy as a critic of art and literature seems to substantiate Habermas" thesis according to which the public sphere originates in the literary world and then expands into a wider political realm. A Cartesian mind-body divide underlines, however, Habermas" evolutionary understanding of rationality. Jonathan M. Hess has recently questioned this opposition between the literary-representational body politic (which belongs to the age of absolutism) and the purely rational discourse of the fully developed bourgeois public sphere. Hess counters Habermas" argumentation by pointing out that the seventeenth and eighteenth century absolutism of the French and German monarchies "claimed its own form of political rationality" and that therefore the Habermasean separation between the rational, on the one hand, and, the body politic, on the other, was not historically water tight (Hess,p. 145).

Hess" critique of Habermas" understanding of the eighteenth century public sphere helps explain Herder"s ambiguous attitude toward literature and art as selfenclosed entities set apart from embodiment-Herder is of course drawn to sculpture because it is a presentation rather than merely representation of the body. Due to his fascination with presentation, Herder was concerned that the world of letters, in particular, and of representation, in general, falls prey to paralysis. He feared that the world of letters would deaden rather than enliven its participants. In order to avoid the potentially paralyzing effect of artistic production and consumption Herder attempted to link the work of intellectual inquiry to the sphere of medicine and the body. In his work as art critic he attempted to connect the apparently irrational realm of the corporeal with the workings of the mind.

Touch or the outdoing of dualisms

In Scupture, in particular, Herder develops an aesthetics that differentiates itself from high levels of ideation. In some strands within the idealist tradition a gulf opens which separates both the body from the mind and ordinary inclinations from the sublime sphere of duty and reason (see [START_REF] Mack | German Idealism and the Jew. The Inner Anti-Semitism of Philosophy and German Jewish Responses[END_REF]. By abolishing these hierarchical divides Herder"s aesthetics is close to Arthur C. Danto"s contemporary philosophy of art. From a historical perspective one could establish the link between Herder and Danto via Hegel. Herder was a major influence on Hegel. Danto"s aesthetics, in turn, owes much to Hegel. Here I am not, however, concerned with historical itineraries but, instead, want to focus on the ways in which Herder shares with Danto"s aesthetics the endeavor to "overcome two boundaries-the boundary between high art and popular art, on the one hand; and the boundary between objects that were works of art and ordinary objects that were parts of daily culture but had not, until then, ordinarily been thought in terms of art" (Danto, 2005, p. x). This is not to say that Herder is a pop art theorist. He of course contemplates not the installations of twentieth century artists but classical Greek sculpture. He approaches, however, classical Greek sculpture in a popular, non-high art way. Here sculpture is not the subject of art criticism but an opening toward a novel and more holistic understanding of human embodiment. His view of humanity is decisively egalitarian: he does not distinguish between artists and ordinary people; nor does he erect boundaries between children and grown ups: indeed as we will see soon, one of the starting points for his argument about touch and sculpture is the very ordinary setting of a nursery.

Clearly Herder theorizes sculpture via the common sense of touch and the common organ of the hand in order to close the gap between mind and body as well as that between art and life. As Danto has pointed out this non-dualistic approach shapes much of the contemporary art scene. Referring to Robert Rauschenberg"s famous 1961 expression that the gap between art and life was to be overcome, Danto writes: " "Overcoming the gap between art and life" had at once the ring of a metaphysical battle cry-like closing the gap between body and mind-and a political slogan promising to abolish privilege." (Danto, 2005, p. x. ) Herder was indeed one of the most radical of early democrats. ii Sculpture seems to be a democratic art form: not only is it frequently dedicated to public places but it also tends to display what we all have in common: the body. Both Danto and Herder focus on embodiment and this appreciation of an embodied from of art leads them to an appraisal of sculpture as the most intriguing artistic genre. In this context it is important to point out that neither Herder nor Danto are materialists, at least not in a narrow sense in which Hobbes or Descartes could be labeled materialists. Herder certainly does not espouse a reduction of every form of life to material mechanisms (as found in Descartes" mechanistic understanding of the body). On the contrary Herder allows for an idealism of sorts, one that is, like his philosophy and theology, non-assertive insofar as it does not privilege dogma, absolute truth, or spiritual existence over and above the fluidity and contingency of everyday, embodied life. Herder is to some extent an idealist because he is fascinated by the transformative power of ideas but the transformation in question here is not superimposed on the material sphere but rather emerges out of it. In the same way Herder"s theological sensibility is not opposed to the secular but rather grows out of an engagement with the mundane. Like Danto"s aesthetics, Herder"s aesthetics is concerned with both content/meaning and its embodiment. "An artwork must have content, that is, it must possess aboutness; and it must embody that content," (Danto, 2001, p. 8) and it is this fusion of content and embodiment that Herder has in mind when he pleads for combining sight (focused on aboutness) and touch (focused on the body).

Most importantly, Herder erases the difference between the aesthetic and the ordinary, when he writes that each child learns by precisely fusing touching with seeing. On the first few pages of Sculpture"s second section, Herder combines his interpretation of art with the advice to his readers to go into an everyday nursery: Go into a nursery and see how the young child who is constantly gathering experience reaches out, grasping, lifting, weighing, touching, and measuring things with both hand and foot, thereby acquiring securely and confidently the most difficult but also the most primary and necessary concepts, such as body, shape, size, space, and distance. These concepts cannot be acquired by teaching or explanation, but only through experience, through exploring and trying things out for oneself. In a few moments the child learns more, and learns it more vividly, more truly and more powerfully, then ten thousand years of mere gaping and verbal explanations could provide. By continually combining his sense of sight with his sense of touch, allowing each to test, extend, enhance, and strengthen the other, he forms his first judgments (Herder, 2002, p.37).

The intellectual task of forming a judgment does not begin in the lofty settings of museums, lecture theatres or laboratories. Instead it originates in a basic nursery.

Significantly the nursery has strong associations with nourishment and primary care.

As such it is related to the corporeal sphere. Herder in fact focuses on the body in his discussion of sculpture. He celebrates sculpture as the most embodied art form. Art that is closest to Descartes" notion of extension (as opposed to Descartes" notion of thought) requires the most embodied aesthetic reception: that of touch.

We may perhaps better grasp Herder"s theory of touch via Jean-Luc Nancy"s recent discussion of body and thought. At first glance the notion "theory" does not seem to be appropriate in this context, because Herder"s thought is seemingly nontheoretical: he appears to distance himself from detached forms of contemplation and observation. Like Nancy, Herder attempts to immerse writing and thinking into what is embodied: "Writing is thinking addressed, thinking sent to the body, sent, that is, to the thing that displaces, estranges it" (Nancy, p. 19). Herder"s idiosyncratic form of theory or writing retrieves, however, the ancient Greek term thigein for touching.

Apropos Gilles Deleuzes"s conception of contemplation without knowledge, Giorigio

Agamben has argued that "this contemplation without knowledge" at times recalls "the Greek conception of theory as not knowledge but touching (thigein)" (Agamben,. Touch sends thought back to the body. This fusion of the thoughtful with what is embodied enables what I call an immanent alterity. My notion of immanent alterity is based on David Jasper"s theological understanding of a coincidentia oppositorum between the immanent and the transcendent, where both are independent while at same time forming a relationship of interdependence. As Jasper has put it: "It is a perfect coincidence of opposites-the transcendent utterly immanent, and the immanent perfectly transcendent" (Jasper,p. 148). This theological interaction between the immanent and the transcendent informs Herder"s aesthetics. The isomorphism of thought and touch characterizes Herder"s non-dogmatic approach towards both philosophy and theology. Thought as touch and touch as thought discovers the extraordinary at the heart of our ordinary immanent existence. This fusion of the thoughtful with the embodied is neither radically materialist nor radically idealist. Rather it resides within the gap dividing these two extremes and keeps crisscrossing their respective boundaries.

Agamben has called "absolute immanence" the tradition of which Deleuze"s philosophy partakes. Agamben distinguishes between the philosophical trajectory of immanence, which starts with Spinoza and then finds its culmination via Nietzsche in Deleuze and Foucault, and that of transcendence which encompasses Kant, Husserl, Levinas, Derrida. I would argue that Herder"s thought may be a beginning of a third philosophical tradition that touches upon the transcendent within the immanent without confusing or conflating one with the other. This tradition may well include Heidegger whom Agamben situates as the only exception to the rule between the two opposed camps of either transcendence or immanence. As has been intimated above, Herder"s philosophy of touch and immanent alterity could be read as a creative reconception of Spinoza"s ethical thought.

Coming back to Herder choosing not an art collection but a nursery as the starting point for his discussion of sculpture, it is significant that Spinoza"s central notion of the conatus (self-preservation) has strong nursery-like or nutritive aspects.

As Spinoza makes clear in his Ethics that the conatus (i.e. self-preservation) depends on "those things" which "are most useful to us" and "which can feed and maintain" us (Spinoza,p. 159). Thought that begins with the nursery and the nutritive enmeshes the ontological within the ethical; rather than opposing one with the other. Agamben has argued that "the most essential character of nutritive life is not simply growth but above all self-preservation;" and Agamben goes on to distinguish this Spinozist theory, which "brings the paradigm of the soul back to the lower scheme of nutritive life," from "the medico-philosophical", which "seeks carefully to distinguish the various faculties of the soul and to regulate human life according to the high canon of the life of the mind." (Agamben,p. 236). According to Herder we need to return to the nursery in order to retrieve a modern sense of classical sculpture. We need to grasp how we have come to able to reflect upon ourselves and our habitat.

The artistic genre for such education is sculpture, because it is through sculpture that we come to perceive the world in an engaged manner (in a manner that is touching, as it were). Here sight and touch are truly combined. Herder clearly believes that classical sculpture is the "nursery" of humanity. Herder does not, however, plead for a return to antiquity. According to Herder, the Greeks privileged the sense of touch, whereas moderns privilege sight. Herder argues for a "modernist" combination of the two senses which would bring about true Enlightenment. Greek sculpture seems to be a nursery that is too one-sided, too much focused on touch: touch, also requires modern sight. The Greek past is incomplete and so is the modern exclusion of the classical sense of touch. The following will discuss how Herder"s historicism is part and parcel of his plea for a combination of both two seemingly opposed senses and two apparently self-exclusive epochs (that of Greek sculpture and that of modern philosophical sight)

The question of classicism and the importance of history

Herder"s historical consciousness makes him avoid a naïve belief in a possible retrieval of a classical past. He advocates not venerating the past but learning from its mistakes. Crucially, historiography compensates for humanity"s instinctual shortcomings. The empirical data that the study of history provides bear an intriguing resemblance to the sensual complexities which are the subject matter of medical inquiry. Herder"s philosophy of the factual does not, however, turn positivistic. The assembly of historical documents is not enough. Rather humanity has to learn from its mistakes in both the present and the past. What the animal does via sensual instinct, the human needs to perform via the reflective activity of cultural critique. Yet reflection does not operate in an ethereal realm. Rather its work relies on empirical data: on sculpture; sight and touch. This is why Herder calls reflection Besonnenheit.

The term Besonnenheit describes the mind"s dependence on the senses. Herder deliberately uses this term, because it combines the bodily with the rational and the sensuous with the intellectual. When we encounter sculptures we are besonnen, because we combine the embodied (touch) with the rational (sight). Sight is the quick, detached, rational sense and touch is the more obscure, slow, incomplete and bodily sense. Herder"s notion Besonnenheit indicates that thinking depends on the senses. This is exactly the overriding topic of Sculpture. Here Herder sets out to "transform his sight into touch, to make his seeing into a form of touching", so that we come to realize that "sight is but an abbreviated form of touch" (Herder, 2002, p. 41). As he makes clear in the quote about the nursery (see above), this slow and incomplete form of seeing is also more truthful than a disembodied type of sight (that of a vision that is disconnected from touch and thus from bodily experience) even though it remains stuck in features of human imperfection: namely incompletion and obscurity as the lack of a full or complete vision.

Related to his attempt in Sculpture to combine the rational with the embodied, the semantic field of Herder"s favorite notion Besonnenheit creates the stage for an immanent alterity where "oppositions cancel each other out" (Herder, 1991, p. 719) Herder"s notion Besonnenheit is Spinozist rather than merely a translation of Locke"s term "reflection", because it depicts the workings of the mind as idea of the body: the corporal and the bodily lose their respective demarcations and as such they traverse back and forth between their respective centers of gravity. Herder"s term "Besonnenheit" certainly refers to Locke"s notion of reflection. Locke"s term does not, however, incorporate a reference to the senses and the body as does Herder"s Besonnenheit. Beiser sees Besonnenheit as merely a borrowing from Locke: "Borrowing a term from Locke, Herder sometimes calls reason "reflection" (Besonnenheit). He chooses this term to refer to man"s characteristic selfconsciousness" (Beiser, 1987, p. 134). Like his notion of "touch" in his discussion of Sculpture, Herder"s term Besonnenheit is not merely cerebral. Herein consists Besonnenheit's Spinozist slant: it is a word that does not exclude the senses but rather incorporates them.

Already in his Journal of my Voyage in the Year 1769 Herder focuses on

Spinoza"s understanding of the mind as the idea of the body. As with Spinoza, Herder stresses the interdependence of reflection and action. Extending and developing Spinoza"s critique of a Cartesian mind body dualism (see Mack, 2010), Herder advances a philosophy of the senses. He does so not only by taking issue with the predominance of the cerebral (sight) over the corporeal (touch) but also by excoriating a hierarchy of sensual perception. The mind interacts with the body in a way similar in which the senses interact amongst themselves:

Moreover we have to make use of all our senses. The sense of touch, for example is dormant in our time, and the eye takes its place, though often only very inadequately.

[…] In general there is no axiom more noteworthy, and almost more often forgotten, than this: without the body, our mind will not function; if the senses are crippled, the mind is crippled too; if the senses are used vigorously and in proper measure, the mind, too, is invigorated (Herder, 1969, p. 83). iii The parallelism of the expression "in action" or "in use" (im Gebrauch) gets lost in English translation. The mind cannot act without a body. Likewise corporal action (Gebrauch) invigorates cerebral activity. Herder thus emphasizes the interdependence of various bodily and mental performances (such as the interdependence of sight and touch). Without the sense of touch, human perception becomes impoverished. Here Herder implicitly introduces an historical aspect into Spinoza"s critique of the mind/body dualism.

How does history enter into this philosophical topic? Through his historical approach Herder naturalizes human creations: they are not eternal; instead they participate in a natural process of decay. This is Herder"s genetic approach (an approach which J. G. Fichte transforms into his science of knowing.): it de-eternalizes human achievements and it removes contemporary preferences and predilections from the central position they might otherwise occupy in the formation of judgments on the aesthetic value and moral accomplishment of a given product or character. In this context Beiser rightly emphasizes the naturalising approach of Herder"s historical or genetic method: "To understand human creations, Herder argues, we must resist the temptation to see them as eternal or anatural. Rather, we must regard them as products of human history" (Beiser, 1987, p. 142).

Around the time at which Herder embarked on his journey of 1769, he started work on his treatise Sculpture. Here he also develops his genetic approach when he discusses how the antique sense of touch made room for the philosophical sense of sight within modernity. As with poetry, sculpture belongs to a past that cannot replace the present. Herder argues that our relationship to Greek sculpture should be one of friendship rather than one of idolatry. As moderns we must be wary of submitting to an idolatrous and thus mindless worship of the past which would destroy our distinctive particularity: "We should treat them [i.e. Greek sculptures] as friends, not idols. Instead of subjugating ourselves to them, we should treat them, as the name suggests, as exemplars that present to us in bodily form the truth of ancient times, making us aware of the proximity and distance between their form of life and our own" (Herder, 2002, p. 61). The study of what has been forgotten establishes both proximity and distance. The sense of touch as understood by the Greeks belongs to a distant age. In the excerpt quoted above, Herder appreciates the cognitive value of this neglected sense. He does not, however, advocate an imitation of its classical connotation. Neither does he plead for a repetition of Greek sculpture within modernity. Instead he modernizes our understanding of touch so that it incorporatesrather opposes or excludes-the more modern rational sense of sight.

The tactile sense has been marginalized due to the progression made by the non-tactile rational sciences (philosophy is a case in point). Herder, however, argues that the senses are not distinct from reason. Following Spinoza, he perceives of the mind as the idea of the body and this is why reason reasons about the senses.

Reflection is this form of reasoning: it is precisely what Herder"s notion Besonnenheit denotes. As Howard Caygill has pointed out, Herder develops an all-inclusive notion of reflection, one that embraces the lowly faculty of touch: "In place of discrete faculties or capacities for thinking, willing, and language, Herder proposes a totality of human powers which structures itself through "reflection"" (Caygill,p. 177). In order to present an accurate account of reflection, Herder has to attend to the intellectual powers of a haptic perception. From the inception of the age of philosophical inquiry onwards, which according to Herder is one-sidedly based on the sense of sight, the capacity of touch has been marginalized and almost completely forgotten. By retrieving this lost sense, Herder in fact revolutionizes the emerging "modern" discipline of aesthetics: "Instead of reducing aesthetics to taste (Geschmack) by legislating judgement, Herder emphasizes the productive discrimination of tasten.

He points to an alignment of beauty, production, and autonomy in the production of proportion which synthesized a view of beauty as embodiment with a notion of culture as the self-cultivation of freedom through reflective judgement. With this position, the epoch of beauty and the police-state is theoretically superseded" (Caygill,p. 183). In this quote Caygill delineates the groundbreaking effect of Herder"s fascination with an almost forgotten relic of the past: with the sense of touch which modern society has deemed insignificant. Caygill clearly shows how this transformation of aesthetics has important political ramification: by dissolving the divide between the cerebral and the corporal, Herder also undermines the hierarchical societal structure that has been advocated by German political theorists of the police state from Sanuel Pufendorf (1632-94) to Christian Wolff (1679Wolff ( -1754)).

In this way the retrieval of the lost sense of touch has the capacity to revolutionize modern aesthetics as well as politics. Herder"s historical imagination is therefore not of antiquarian interest only. By dint of the historical imagination, the distant past can be rendered relevant for an accurate understanding of contemporary concerns. As long as this proximity does not destroy an awareness of the present the so conceived closeness of the past still remains distant. Herder establishes a fine balance between the ancients and the moderns as well as between the corporeal and the cerebral. He thus does not play off touch against eyesight. Rather, he argues for a combination of both senses (as well as for the non-exclusion of the moderns by the ancients and vice versa), which in turn would enhance rather diminish the life of the mind. Like Greek antiquity the sense of touch belongs to a pre-modern age that is analogous to the ontogenetic stage of childhood. A return to antiquity is thus a return to the nursery. The difference between child and adulthood might indicate the construction of a hierarchy in which one is inferior to the other. This is, however, not Herder"s point. Rather he wants to bring about the anamnesis of that which has been forgotten and repressed p.360). In a way similar to which the divergence between body and mind establishes their mutual interdependence, the present cannot properly function without being cognizant of the past.

Herder neither devalues natural or historical data that constitute the empirical body of knowledge. In his Journal of my Voyage in the Year 1769, however, he dismisses as romantic any belief in the immediacy of the past:

In every age-though in each in a different way-the human race has had happiness as its objective; we in our own times are misled if, like Rousseau, we extol ages which no longer, exist and never did exist, if we make ourselves miserable by painting romantic pictures of these ages to the disparagement of our own, instead of finding enjoyment in the present. Seek then even in biblical times only that religion and virtue, those examples, those forms of happiness, which are appropriate to us: become a preacher of virtue of your age! (Herder, 1969, p. 89) Herder anticipates Nietzsche"s critique of historicism in the essay "The Uses and Abuses of History" that is part of the book Untimely Observations (1873-76).

Crucially, Herder advances this analysis while questioning Rousseau"s one-sided depiction of the "noble savage". This assessment of premodern culture constructs a radical divide between past and present. While acknowledging the difference between the bygone and the contemporary, Herder attempts to save what has passed from being forgotten and dead. He does not advocate an uncritical study of history. Rather, he tries to focus on those elements which are relevant for problem solving in the present. In this way the study of history (Greek sculpture that privileges touch) always already requires the mediation of the present (the modern dominance of sight over touch). Contrary to the common perception of Herder as a romantic, he in fact clearly articulates that humanity cannot find enjoyment in an immediate manner. The senses depend on the reflection of the mind (sight or philosophy) in the same way as the reflection of the mind depends on the work of the senses (touch and embodiment).

Sustainable enjoyment can only be attained through the interaction between these two disparate faculties.

The study of both natural and human history thus enables survival of the senses within the here and now. In his Essay on the Origin of Language Herder dialectically affirms that the present incorporates the non-presence of the past. In the same way as it cannot immediately live in the sensory realm of the body alone, humanity equally cannot do without the reflection on its history, if it wants to survive within the present. Like Spinoza, Herder is concerned not with ideals but with the immanence and the physicality of the conatus. Here, however, life"s ongoing selfpreservation (conatus) pertains not only to the materiality of nature but also to the remnants of the human past (Greek Sculpture). The past represents a cynosure by which humanity can orient itself on its journey into the future.

Thinking as Touching

What is crucial here is that Herder not only attempts to excavate a rather neglected sense (touch) but that he argues that without this rather marginalized faculty we would be unable to mature intellectually. In a highly ironic and brilliant move Herder argues that without touch we would not be able to see. Herder reverses Denis Diderot"s question (in his Letter on the Blind) "What can the blind perceive without the use of touch" by asking "What could the sighted perceive without the use of touch?" (Gaiger,p. 15). His reply is quite clear: the sighted are blinder than the blind when they too much trust their eyes. The exclusive reliance on eyesight is not to be trusted, because our sense of sight depends on light. This becomes amply clear in the dark when we have to rely on touching things in order to orient ourselves. Sight, Herder argues, only provides us with the shadow or contours of things, whereas touch uncovers the obscure but solid truth of our embodied world.

By interpreting light and sight not so much as agents of Enlightenment but, on the contrary, by associating the sunlight and what it enables, namely eyesight, with shadows of the truth, Herder implicitly reverses Plato"s famous parable where the inhabitants of the cave are separated from the real sunlight by the walls of the cave and thus only see darkly shadows rather than the truth itself. Herder not only reverses Denis Diderot (as has been pointed out by Jason Gaiger in the quote above), and Plato but also the godfather of modern scientific reason, Descartes. Qualifying Descartes"s privileging of the mind over and above the body, Herder advances a novel account of humanity"s uniqueness. Humanity distinguishes itself from other animals not only through its rational capacity but equally through a physiological one: that of touching the embodied world with its hands (later on, namely in the Ideas, Herder defines humanity"s uniqueness as not only based in its mental capacities but also in its physiology when he says that what is singular about humans is not its reasoning as such but its upright position). Countering the strong ideational tendencies in Plato and Descartes, Herder argues that we are only capable of conceiving ideas because we are capable of touching bodies with our hands:

The light that strikes my eye can no more give me access to concepts such as solidity, hardness, softness, smoothness, form, shape, or volume than my mind can generate embodied, living concepts of independent thinking. Birds, horses, and fish do not possess these concepts. Only human beings have them, because alongside reason we possess a hand that can feel and grasp. If we did not have this, if we had no means by which we could confirm the existence of a body for ourselves through our own bodily feeling, we could only infer and guess and dream and fabricate, and we could know nothing for certain. (Herder, 2002, p. 36) The mind as self-enclosed entity is liable to fall prey to a world of dreams and fabrications that distort rather than grasp the world (this is of course the very subject matter of the pre-critical Kant"s brilliant Dreams of a Ghost Seer of 1766-roughly the time during which Herder composed Sculpture). In order to form concepts by which we can orient ourselves in the world we need to traverse back and forth between the cerebral and the bodily. The certainty for which Cartesian rationalism strives can ironically only be attained by abandoning Descartes"s mind-body divide.

Sculpture is the genre which helps coordinate our cerebral capacities with the tangible sphere of touch and embodiment. According to Herder sculpture demands of us an endless journey from mind to body and vice versa. This mobility characterizes sculpture"s modernity. Unlike a more pre-modern, exclusive reliance on touch, Herder"s philosophy of touch does not diminish or belittle the importance of rational detachment. As we have seen, Herder advocates a fusion of both sight and touch. This interdependence of the senses-which his analysis of sculpture illuminates-runs parallel to his dialogical approach toward historical and cultural diversity. The incomplete state of our capacities as well as their diversity requires a mobility that traverses fields of strengths which seem to be disconnected from each other. According to Herder, we-being constituted by diverse mental and bodily abilities-are, however, all interconnected in so far as we share one common creator-however different our conceptions of this creator-God are (be it Muslim, Jew, Christian, Hindu etc.). By interconnecting sight and touch, medicine and the humanities as well as the ancient and the modern, sculpture outdoes various dualities and thus holds out the promise of not only a co-existence but, more importantly, a mutual preservation of what is diverse.

* This article is related to but separate from the discussion about the hidden Enlightenment of diversity as advanced in my forthcoming book Spinoza and the Specters of Modernity: the hidden Enlightenment of Diversity fromSpinoza to Freud, (New York: Continuum, 2010).


Jonas Fischer
email: jonas.fischer@ovgu.de

Jens Strackeljan

A nonlinear numerical simulation of a lab centrifuge with internal damping

Keywords: rotordynamics, internal damping, nonlinearity, rotor shaft joint

This paper is about numerical simulations of dissipation processes in rotor shaft joints of rotor systems. Based on measurement results a nonlinear simulation model of a lab centrifuge is stated. The effects of internal damping in combination with nonlinear stiffness and friction in the rotor shaft joint of the lab centrifuge are worked out. It is shown that the nonlinearities cause the amplitudes to rest limited once increased amplitudes due to internal damping appear. One focus is the derivation of suitable force laws describing the mechanisms of the components within the connection.

Introduction

Particularly in the field of lab centrifuges problems like rough running, high excitations of the rotor or even rub between housing and rotor can occur in the post critical speed range. The problems are sometimes traced back to internal damping. The word internal means, that damping and friction act in the rotating part of the rotor system. Internal damping influences the stability of the rotor system in the post critical speed range significantly.

Often the joint between rotor and shaft is the main cause for internal damping and internal friction. The mechanisms of energy dissipation within the connection are in many cases nonlinear making a simulation demanding. Nevertheless a simulation remains worthwhile as it assists to analyze the reasons of internal damping and to find measures to enlarge the stability margins.

In the present article a lab centrifuge is examined where high rotor deflections due to internal friction are observed above a certain speed of revolution. Thereby the occurrence of high rotor deflections and rough running depends decisively on the height of the systems excitation by an impact. A sketch of the lab centrifuge is shown in figure 1. The rotor shaft joint is designed tool free with a plug and lock mechanism. It enables an easy change of rotors as this single centrifuge runs with several different rotors. The connection between rotor and shaft consists of three parts: a clamp used to lock the connection, an o-ring which is used to center the rotor and a plastics. The plastics serves as guidance for the rotor.

In figure 2 and 3 the accelerations which were measured at the motor base of the lab centrifuge are shown. The unit was operating at a frequency of revolution of 125 Hz. An additional excitation was performed with an impact hammer hitting the motor base. Depending in the height of the excitation increased amplitudes or a damped oscillation could be observed. The increased oscillation is bounded. One can assume that the clamp opens if the force in the connection clamp is higher than the friction force. This is the case, if the impact at the motor base is high. Thus the internal damping of the o-ring starts to work. From literature it is known, that linear internal damping causes the amplitudes to rise unbounded ( [START_REF] Gasch | Rotordynamik[END_REF]). As the amplitudes are bounded in figure 2, the force laws for o-ring and plastics have to be nonlinear. The stiffness of o-ring and plastics are nonlinear in this case due to the clearance between rotor and shaft.

A nonlinear simulation was performed to analyze the mechanisms of the described phenomenon exactly and to identify the main parameters influencing the stability of this system. The results are depicted in section 4.1. In section 2 the simulation model of the lab centrifuge and the nonlinear model of the rotor shaft joint is explained. A linear eigenvalue simulation is used to evaluate the simulation parameters. The results can be found in section 3.

Simulation model

In figure 4 one can see a sketch of the simulation model. It consists of an elastic shaft, a rigid rotor and three linear spring-damper-elements describing the ball bearings between stator and shaft and the rubber bearing. The connection between rotor and shaft is modelled by three nonlinear elements. The dynamic behaviour of the rotor system is described by the discrete differential equation in the inertial coordinate system. [START_REF] Gasch | Rotordynamik[END_REF] with stiffness matrix [K], damping matrix [C], gyroscopic matrix [G] and mass matrix [M ]. Vector {D} consists of the nodal displacements, vector {F ext } of the linear external and {F nl } of the non-linear internal and external forces. To obtain equation ( 1), the principle of linear momentum is first stated in a body fixed coordinate system and subsequently transformed into the inertial coordinate system. The internal forces

CLAMP

[M ]{ D}+[[C]+[G]]{ Ḋ}+[K]{D} = {F ext }+{F nl },
{F int } * = [K] * •{D} * + [C] * •{ Ḋ} *
describing the stiffness and damping properties of the shaft and the linear part of the shaft rotor connection in a coordinate system rotating with the frequency of revolution are transformed into the inertial coordinate system. The * denotes that the entity is used in the rotating coordinate system.

The shaft of the rotor system is discretised by 8 beam elements, including transverse shear deformation, according to Timoshenkos beam theory ( [START_REF] Cowper | The shear coefficient in timoshenko's beam theory[END_REF]). The upper and lower shaft bearing of the centrifuge and the nodes of the rotor are coupled rigidly. Constraints are used to connect the movements of the corresponding nodes.

The parameters to state the stiffness and mass matrix of the shaft follow directly from the shafts geometry and material properties. Linear stiffness and damping coefficients of the rubber support are deduced by separate tests. The inertia moments of the rotor are obtained numerically.

Simulation model of the rotor shaft joint

On the one hand the mechanisms of damping which occur in the rotor shaft joint can be traced back to material damping. This is for example the case for the o-ring connecting rotor and shaft. Another mechanism are micromovements and as a result friction between the single components of the rotor shaft connection.

The energy dissipation of material damping is describable by viscous internal damping. The Voigt-Model can be used for simulation. In [START_REF] Shaw | Instabilities and bifurcations in a rotating shaft[END_REF] a linear viscous internal damping force is deduced from this material model. The force law is extended by a nonlinear part if huge amplitudes are considered. In the linear case, one can expect unbounded amplitudes above a certain frequency of revolution. In [START_REF] Genin | Stability in the three dimensional whirling problem[END_REF] a similar nonlinear force law is applied. It is shown that the analytical solution of the considered rotor system is bounded for all frequencies.

A fundamental analyze of rotating dry friction can be found in [START_REF] Tondl | Some Problems of Rotor Dynamics[END_REF]. The analytical steady state solution and an investigation about the stability of the solution is accomplished by measurement results.

Newer works principally deal with the description of micromovements and the resulting structural damping in practical applications.

There exists different approaches to model friction between the single components of a connection.

By one hand measured hysteresis are described by phenomenological models, like for example the Dahl-Model [START_REF] Dahl | A solid friction model[END_REF]. This approach does not allow to predict directly the motion of the rotor system. Parameters are determined for each component of a device, for which hysteresis loops were measured. The parameters are subsequently used for a description of the whole device.

On the other hand there exist micro slip models like for example the model of Masing or of Menq [START_REF] Menq | The influence of a variable normal load on the forced vibration of a frictionally damped structure[END_REF] or models, which take into account the tip elasticity of the coarsens of the contact area like in [START_REF] Shoukry | A mathematical model for the stiffness of fixed joints between machine parts[END_REF] and [START_REF] Burdekin | An elastic mechanism for the microsliding characteristics between contacting surfaces[END_REF]. Micro slip means that in the beginning only one part of the contact area is slipping prior the entire contact area slips.

In rotor dynamics micro slip models are used since some years especially for modelling the processes between turbine blades and shaft respectively the processes in spline shafts. In [START_REF] Wettergren | Material and microslip damping in a rotor taking gravity and anisotropic bearings into account[END_REF] the dissipation energy caused by micro slip and material damping in slot wedges in a turbine generator are analyzed.

Furthermore it is possible to simulate the connection with contact elements of the finite element method instead of the use of micro slip models. In [START_REF] Charleux | Numerical and experimental study of friction damping in blade attachments of rotating bladed disks[END_REF] a technique is proposed to calculate the nonlinear response of an harmonic excited turbomachine at constant speed of revolution. Blades and rotorshaft are generated as finite element model. Contact elements are used for the joint. The simulation is done by the DLFT-method (Dynamical Lagrange Frequency Time Method, [START_REF] Nacivet | A dynamic lagrangian frequency-time method for the vibration of dry-friction-damped systems[END_REF]), assuming that only periodic forces and deflections occur. Measurements show a good agreement with the simulation results. But there is no possibility to take into account arbitrary excitations, like impact forces or to compute a run-up of the rotor system.

As simulation approach for the considered lab centrifuge, the rotor shaft joint is divided in parts which are different in construction. For every part of the joint a nonlinear force law is stated. Parameters which are necessary to describe the mechanisms of every component of the connection are determined in separate tests. This is done for example by recording hysteresis in a hydro-puls test rig. Thereby one obtains a connection model which is designed comparatively easy and which is suitable for transient calculations. The model is sufficiently exact to determine the influence of the single mechanisms.

The rotor shaft connection is modelled with three nonlinear elements. The choice of parameters and of force laws is explained in the following.

Nonlinear force law clamp

If the force in the connection is higher than the friction force, the clamp opens and dry friction occurs. The rigid link between shaft node and rotor node can be implemented in the simulation model in different ways. The nodes can be linked by lagrange multipliers or by penalty-functions. If lagrange multipliers are used, an implicit time integration method has to be used for solving the equation of motion. As the model is nonlinear, a Newton iteration in each time step is necessary to obtain the nonlinear forces. The penalty method enables the use of an explicit integration method or a linear implicit direct integration method. But it is sometimes difficult to find appropriate penalty values. if the clamp is closed the motion of shaft node (node S in figure 5) and rotor node (node RS) is coupled directly by constraints. Node RS is a slave node which is connected rigidly with the rotor master node RM in a distance of l bc . Hence the equation coupling the motion of the rotor master node and the shaft node S by the constraint matrix [ tc ] is:

v S w S {dS } = 1 0 • • • 0 l bc 0 1 • • • -l bc 0 [ tc] ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ v RM w RM . . . θ yRM θ zRM ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ {dRM } (2)
The entities with index S are the deflections of the shaft node and the ones with index RM are the deflection of the rotor master node. If we consider the reduced deflection vector {D r } of the whole system, containing only the master node deflections and the deflection vector {D} of the non coupled system, we get the constraint matrix

[T c ]. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ . . . {d S } . . . {d RM } . . . ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ {D} = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ • • • • • • • • • [0] [ tc ] [0] • • • • • • • • • [0] [I] [0] • • • • • • • • • ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ [Tc] ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ . . . {d RM } . . . ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ {Dr } (3)
The number of degrees of freedom of the whole system is reduced by two if the clamp is closed. The equation of motion is transformed completely by the transformation matrix [T c ]. As the clamp opens the amount of degrees of freedom changes.

After each time step of the numerical integration, the state of the clamp is checked. If the force in the closed connection F * cc is higher than the maximum adhesive force F s , the clamp opens and a nonlinear dry friction force F * c acts between both nodes (equation 5). The sticking condition is

|F * cc | < F s = μ 0 • F * N (4)
In the simulation a friction coefficient μ 0 = 0.8 is used, according to the combination steel -steel ( [START_REF]Dubbel, Taschenbuch für den Maschinenbau[END_REF]). The normal force F * N is obtained from the preliminary tension force of the spring element. Equation (5) describes the dry friction force F * c in the rotating coordinate system.

F * c = -μ • F * N • Δ Ḋ * |Δ Ḋ * | (5)
The force F * c is transformed into the inertial coordinate system. The friction coefficient in sliding state μ is assumed to be 5% smaller than μ 0 in the sticking condition. The variable Δ Ḋ * denotes the relative velocity between rotor node and shaft node, expressed in the rotating coordinate system.

Nonlinear force law o-ring

In order to find a force law, which describes the properties of the o-ring the test configuration shown in figure 6 was build. The test configuration with two o-rings was analyzed in a hydro pulse unit. The rotor was charged with a harmonic force. The frequency was 1.0 Hz and the amplitude of the force was 23.75 N . The clearance between the shaft and the rotor was the same clearance like in the rotor shaft connection of the lab centrifuge. A capacitive sensor was used to measure the displacement between rotor and shaft. The displacement is plotted in figure 7. The measured hysteresis loop flattens for high displacements as the clearance between rotor and shaft is reached. In addition a simulation of the test was performed nu- merically. The simulation enables to deduce appropriate values for the parameters of different force laws.

The use of a linear force law for the o-ring (equation ( 6)) leads to an unacceptable difference between measurement and simulation.

F o = c i • Δ ẋ + k i • Δx (6)
The linear damping coefficient of the o-ring is c i , k i is the stiffness of the o-ring and Δx is the relative displacement between shaft and rotor.

A cubic o-ring model reflects the stiffening of the oring at higher forces (equation ( 7)). The force deflection function remains smooth such that the numerical implementation does not provoke any difficulties. A cubic stiffness k inl is used.

F o = c i • Δ ẋ + k i • Δx + k inl • Δx 3 (7)
However this model has some drawbacks. The linear stiffness is to high and the bending of the curve is not reflected very well. For a numerical simulation of the rotor system, a bilinear cubic force law can be used (equation ( 8)).

F o = c i • Δ ẋ + k i • Δx + k ir • (Δx -s) + k inl • Δx 3 (8)
A new linear relative stiffness k ir is introduced which is multiplied by the difference between relative displacement Δx and clearance s and which only acts, if the relative displacement is higher than the clearance. The linear relative stiffness is much higher than the linear stiffness. The simulated and measured hysteresis loop coincide well.

Nonlinear force law plastics

The plastics is modelled similar to the o-ring, but with lower internal damping c i and a higher stiffness k i and k inl . As the stiffness of the plastics is much higher than that of the o-ring, it is not necessary to use a relative stiffness in the simulation. Hence the force law of equation ( 7) is applied.

Linear simulation of the lab centrifuge

The simulation model consists of many different parameters like stiffness properties, mass properties and damping properties. It is expensive or sometimes impossible to determine all parameters exactly. To evaluate the used parameters and to adjust them to measurement results, a linear simulation of eigenvalues and damping values was performed. The eigenvalues were measured with a closed clamp.

If the clamp of the rotor shaft joint is closed one can assume that the system is linear. Hence for the simulation of eigenvalues all nonlinear effects of the rotor shaft joint were neglected and the clamp was closed.

In figure 8 the measured and calculated eigenvalues are shown.

In the linear case the damping value of the second forward whirl is important for the stability of the system. If it becomes negative the amplitude of the second forward whirl rise unbounded. The damping values corresponding to the second forward whirl of figure 8 are depicted in figure 9 amplitudes below 180 Hz, one has to consider another mechanism, the opening of the clamp. It is not possible to describe the effect shown in figure 2 with a linear simulation model.

Nonlinear simulation of the lab centrifuge

To calculate the release of the clamp the model used for the calculation of eigenvalues was enlarged with the nonlinear properties of the rotor shaft connection. A modified Rosenbrock method was used to perform the simulation in time domain. Rosenbrock methods are applicable for nonlinear systems. They are best suited for systems of stiff differential equations as explicit integration methods perform nonsatisfyingly for those problems. The algorithms used for the simulation is a linear implicit 2nd and 3rd order Runge-Kutta method with step control. After each time step the state of the clamp is checked with the condition of equation ( 4). If the state of the clamp turned out to be different from the state which was assumed for the calculation in the time step, this time step is repeated with the new state of the clamp.

Results of the nonlinear simulation

The simulation was performed in a first step at a frequency of revolution of 125 Hz. The unbalance load was adjusted to the oscillation measured prior to the release of the connection. The mass of unbalance is 0.25 g and the radius of unbalance is 109.34 mm.

The height and the duration of the impact acting on the motor base were adjusted to the measured impact (figure 10 and figure 12). After the first impact in the test the motor base starts to oscillate. The impact hammer is not moved away fast enough. Hence it bounces against the motor base again. This explains the oscillations which could be observed in the measurement in figure 10 and figure 12.

In figure 11 and figure 13 the computed and measured accelerations of the motor base are depicted. Simulation and measurement results coincide well.

The simulation reflects also the phenomenon that increased amplitudes like in figure 13 occur only if the impact force is high enough (figure 12). If the impact is lower (figure 10), the clamp stays closed and the amplitudes diminish (figure 11). vibrations occur, the highest forward whirl below of the frequency of revolution of 125 Hz turned out to be the most dominant peak in the frequency spectrum (figure 14). This result holds for measurement and simulation. The highest forward whirl at this speed of revolution is 39.2 Hz (measurement) and 41.0 Hz (simulation).

It agrees with results from analytical research in internal damping. If the internal damping is of linear viscous damping character, it is known that the damping values of the forward whirls can get negative such that the frequency of the self-excited vibration is the same like the corresponding forward whirl (e.g. [START_REF] Fischer | Stability analysis of high speed lab centrifuges considering internal damping in rotor-shaft joints[END_REF]).

In [START_REF] Ishida | Forced oscillations of a rotating shaft with nonlinear spring characteristics and internal damping[END_REF] a two degrees of freedom rotor with internal dry friction is analyzed analytically. It is denoted, that in the post critical speed range a bounded self-excited oscillation with the frequency of the first forward whirl occurs. In our case the same kind of dry friction is working in the clamp once the connection is released.

As the rotor shaft connection is stiffer, if the clamp is closed, the frequency value of the considered forward whirl is higher (figure 15, 42.4 Hz (measurement) and 42.0 Hz (simulation)). There is still positive damping such that the frequency is not dominating in the RMS spectrum in the same manner like for the open clamp. Because of the nonlinearities in the force laws of o-ring and plastics the amplitudes for the case of an open clamp are bounded. Curve A in figure 16 shows the magnitudes of the motor base accelerations. If we neglect the relative stiffness k ir of the o-ring in equation [START_REF] Shoukry | A mathematical model for the stiffness of fixed joints between machine parts[END_REF] we obtain the force law of equation [START_REF] Menq | The influence of a variable normal load on the forced vibration of a frictionally damped structure[END_REF]. Now only cubic nonlinearities are taken into account. This corresponds to curve B (figure 16). In curve C the cubic stiffness k inl of o-ring and plastics is reduced to 10 % of the original value and in curve D to 1 % of the original value. It is obvious that the amplitudes of the motor base increase the smaller the nonlinearities of the joint stiffness are. In figure 17 measurement and simulation results at a frequency of revolution of 150 Hz are shown. The height of the impact force is 230 N in measurement and simulation.

A run-up and a run-out is depicted in figure 18. The same radius of unbalance like in figure 14 and 15 is used but the unbalance mass is augmented to 10 g.

Conclusion

The description of different parts of the rotor shaft joint of a lab centrifuge by different nonlinear force laws has proved to be a promising way in simulation. The nonlinear simulation results and related measurements are in good agreement. It is possible to show the influence of internal damping in combination with nonlinear stiffness and dry friction to the behaviour of the device numerically. The nonlinearity of the rotor shaft connection turned out to be responsable for the limitation of the amplitudes once the internal damping of the connection starts to work in the post critical speed range. Measurement and simulation results coincide with results which are known from analytical models with internal damping.

Fig. 1

 1 Fig. 1 Schematic model of the lab centrifuge (without safety containement and housing).

2 Fig. 2 2 Fig. 3

 2223 Fig. 2 Measured accelerations at the motor base after a high excitation.

Fig. 4

 4 Fig.[START_REF] Genin | Stability in the three dimensional whirling problem[END_REF] Simulation model of the lab centrifuge. Coordinate system x, y, z with displacements u, v, w (translation) and θx, θy, θz, (rotation).

ForFig. 5

 5 Fig. 5 Constraint model clamp.

FFig. 6

 6 Fig.[START_REF] Dahl | A solid friction model[END_REF] Test configuration with two o-rings analyzed in a hydro pulse unit.

Fig. 7

 7 Fig. 7 Hysteresis loop of the test configuration with two o-rings, simulation and measurement.

  . If the clamp is closed, both the measured and the simulated damping are positive up to a speed of revolution of 180 Hz. Hence, to obtain rising

Fig. 8

 8 Fig.[START_REF] Shoukry | A mathematical model for the stiffness of fixed joints between machine parts[END_REF] Forward whirls (fw) and backward whirls (bw) of the linear simulation model (solid and dashed lines) compared to measurements (tokens).

Fig. 9

 9 Fig. 9 Damping values of the second forward whirl. Linear simulation model compared to measurements.

Fig. 10

 10 Fig. 10 Low impact force in measurement and simulation at a frequency of revolution of 125 Hz, connection clamp stays closed.

Fig. 11 Fig. 12 Fig. 13 Fig. 14

 11121314 Fig. 11 Measured and computed acceleration of motor base at a frequency of revolution of 125 Hz, low impact force.

Fig. 15

 15 Fig. 15 RMS spectrum of time signal of figure 11 at a frequency of revolution of 125 Hz, connection clamp closed.

Fig. 16

 16 Fig. 16 Magnitude of motor base accelerations with varied joint parameters: A: original parameters like in figure 13, B-D: force law of equation (7) for o-ring and plastics, C: 10% cubic stiffness k inl of B, D: 1% cubic stiffness k inl of B.

  Fig. 17Measured and computed acceleration of motor base, high impact force at a frequency of revolution of 150 Hz.

Fig. 18

 18 Fig. 17Measured and computed acceleration of motor base, high impact force at a frequency of revolution of 150 Hz.

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