Mark Van Atten
email: vanattenmark@gmail.com

A note on Leibniz's argument against infinite wholes

Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. 1

1. The part-whole axiom is also referred to as the 'Aristotelian principle' (e.g., [START_REF] Benci | An Aristotelian notion of size[END_REF] or 'Euclid's Axiom'. The former label is justified as it follows from what Aristotle says at Metaphysics 1021a4: 'That which exceeds, in relation to that which is exceeded, is "so much" plus something more' (Aristotle 1933, 263); the latter, to the extent that it figures as Common Notion 5 in Book I of Euclid's Elements from (at the latest) Proclus on (Euclid 1956, 232). See also Leibniz's 'Demonstratio Axiomatum Euclidis' (1679), Leibniz, 1923-, 6,4:167. 2. This contributes to showing that there is no intrinsic obstacle in Leibniz's philosophy to combining it with Cantorian set theory, as Kurt Gödel wished to do. For further details on this aspect of Gödel's thought, which provided the motivation for writing the present note, see van Atten 2009. Neither Gödel's published papers, nor, as far as I can tell from the currently existing partial transcriptions from Gabelsberger shorthand, his notebooks contain a direct comment on Leibniz's argument. However, in his paper on Russell from 1944, he wrote:

Nor is it self-contradictory that a proper part should be identical (not merely equal) to the whole, as is seen in the case of structures in the abstract sense. The structure of the series of integers, e.g., contains itself as a proper part. (Gödel 1944, 139) Among other things, Gödel says here that it is consistent that an equality relation holds between a proper part and the whole. This entails a rejection of Leibniz's argument. 1 2. This refutation does not presuppose the consistency of Cantorian set theory.

3. This refutation does not cast doubt on the part-whole axiom.

A note on sources: although Leibniz's texts used below range from 1672 to after 1714, they all express the same view on the issue at hand. Of each the year will be given where possible. Emphasis in quotations from Leibniz is his. Translations without a reference are mine.

Leibniz's argument and its refutation

The following presentation of Leibniz's argument and its refutation is meant to establish claim (1) and to set the stage for the defence of claims ( 2) and (3).

Leibniz denied the existence of infinite wholes of any kind. 3 For example, while he acknowledges that there are infinitely many numbers For it cannot be denied that the natures of all possible numbers are really given, at least in God's understanding, and that as a consequence the multitude of numbers is infinite. 4 he denies that this multitude forms a whole: I concede [the existence of] an infinite multitude, but this multitude forms neither a number nor one whole. It only means that there are more terms than can be designated by a number; just as there is for instance a multitude or complex of all numbers; but this multitude 3. Friedman 1975, 338 suggests that even so, Leibniz might have been willing to accept the for him inconsistent concept of an infinite whole as a fiction that may prove useful in calculations, on a par with his acceptance of imaginary roots in algebra. To illustrate this point, Friedman refers to [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz ( ) 1882, 145, 145. See also Leibniz's letter to Des Bosses of 1 September 1706: 'properly speaking, an infinity consisting of parts is neither one nor a whole, and can only be conceived of as a quantity by a mental fiction.' ('proprie loquendo, infinitum ex partibus constans neque unum esse neque totum, nec nisi per fictionem mentis concipi ut quantitatem', Leibniz 1875, 2:314) 4. 'Neque enim negari potest, omnium numerorum possibilium naturas revera dari, saltem in divina mente, adeoque numerorum multitudinem esse infinitam.' (Leibniz 1875, Leibniz to Des Bosses, 11/17 March 1706;2:305) is neither a number nor one whole. (Leibniz 1849, Leibniz to Joh. Bernoulli, 21 February 1699; 3:575) 5,6 The point of departure for Leibniz's argument that an infinity cannot be a whole is the part-whole axiom, which he justifies by the following definitions and argument: 7

If a part of one thing is equal to the whole of another, the former is called greater, the latter less. Hence the whole is greater than a part. For let the whole be A, the part B. Then A is greater than B, because a part of A (namely, B) is equal to the whole of B. This can be expressed in a syllogism whose major proposition is a definition, its minor an identity:

Whatever is equal to a part of A is less than A, by definition. But B is equal to a part of A (namely, to B), by hypothesis. Therefore B is less than A. [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF](Leibniz , 668 (after 1714)) 8,9 5. 'Concedo multitudinem infinitam, sed haec multitudo non facit numerum seu unum totum; nec aliud significat, quam plures esse terminos, quam numero designari possint, prorsus quemqdmodum datur multitudino seu complexus omnium numerorum; sed haec multitudo non est numerus, nec unum totum.' 6. Also: 'PH. … Nothing is clearer than the absurdity of an actual idea of an infinite number. TH. I agree. But this is not because one couldn't have the idea of the infinite, but because an infinity cannot be a true whole' ('PH. … rien n'est plus sensible que l'absurdité d'une idée actuelle d'un nombre infini. TH. Je suis du même avis. Mais ce n'est pas parcequ'on ne sauroit avoir l'idée de l'infini, mais parcequ'un infini ne sauroit estre un vrai tout', [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz ( ) 1882, 146) , 146) 7. Although an axiom is a proposition that is evident, Leibniz sees two uses for a demonstration, that is, a reduction to A=A: it contributes to the unification of the sciences and to the analysis of ideas. See Couturat 1901, 200ff. 8. 'Si pars unius sit aequalis alteri toti, illud vocatur Minus, hoc Majus. Itaeque Totum est majus parte. Sit totum A, pars B, dico A esse majus quam B, quia pars ipsius A (nempe B) aequatur toti B. Res etiam Syllogismo exponi potest, cujus Major propositio est definitio, Minor propositio est identica: Quicquid ipsius Q parti aequale est, id ipso A minus est, ex definitione, B est aequale parti ipsius A, nempe sibi, ex hypothesi, ergo B est minus ipso A.' (Leibniz 1849, 7:20) 9. See also [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF], 518/Leibniz 1969, 267 (around 1686); [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6/4:167 (1679)); Leibniz 1875, 7:300;[START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF], 7:274 (1695, according to de Risi 2007, 82); Leib-

The major proposition shows that for Leibniz, 'part' means 'proper part', i.e., a part which is not equal to the whole. His argument from this axiom against infinite wholes runs as follows: 10

There is no maximum in things, or what is the same thing, the infinite number of all unities is not one whole, but is comparable to nothing. For if the infinite number of all unities, or what is the same thing, the infinite number of all numbers, is a whole, it will follow that one of its parts is equal to it; which is absurd. I will show the force of this consequence as follows. The number of all square numbers is a part of the number of all numbers: but any number is the root of some square number, for if it is multiplied into itself, it makes a square number. But the same number cannot be the root of different squares, nor can the same square have different roots. Therefore there are as many numbers as there are square numbers, that is, the number of numbers is equal to the number of squares, the whole to the part, which is absurd. (Leibniz 2001, 13) 11 The reductio argument that Leibniz presents here can be reconstructed as follows:

1. The infinite multitude of the numbers forms a whole. (Assumption) 2. Every square is a number, but not vice versa. (Premise) 3. The multitude of the squares is equal to a part of the whole of the numbers. (1, 2) niz 1849, 3:322 (1696). 10. Leibniz presented the same argument on various occasions: see Leibniz, 1923-, 2,1:226 and228 (1672); [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 4,3:403 (1675)); [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6,3:463 (1675)); [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6,3:168 (1676)); [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6,3:550-3 (1676)); his draft letter to Malebranche of 22 June 1679, Leibniz 1875, 1:338. See also the reference to it in a letter to Johann Bernoulli of 1698, Leibniz 1849, 3:535. 11. 'Nullum datur Maximum in rebus, vel quod idem est Numerus infinitus omnium unitatum non est unum totum, sed nihilo aequiparatur. Nam si numerus infinitus omnium unitatum, seu quod idem est, Numerus infinitus omnium numerorum, sequetur aliquam eius partem esse ipsi aequalem. Quod est absurdum. Consequentiae vim ita ostendo. Numerus omnium Numerorum Quadratorum est pars Numeri omnium Numerorum: at quilibet Numerus est radix alicuius Numeri quadrati, nam si in se ducatur, fiet aliquis numerus quadratus; nec idem numerus potest esse radix diversorum quadratorum, nec idem quadratus diversarum radicum, tot ergo sunt Numeri, quot Numeri quadrati, seu Numerus Quadratorum aequalis est Numeri Numerorum, totum parti, quod est absurdum.' [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF](Leibniz, -, 6,3:98 (1672-3)) -3)) 4. There exists a bijection between the multitude of the numbers and the multitude of the squares. (Premise)

5. The multitude of the squares is equal to the whole of the numbers.

(1, 4) 6. A part of the whole of the numbers is equal to the whole of the numbers. (3, 5)

7. The whole is greater than its parts. (Premise) 8. Contradiction. (6, 7)

9. Therefore, the infinite number of all numbers do not form a whole.

(1, 8)

Leibniz holds the propositions in lines 2, 4 and 7 to be true; the source of the contradiction that arises in line 8 for him is the assumption made in line 1.

Russell and others have observed that Leibniz's argument is not correct because it rests on an equivocation on the concept of equality. 12 Clearly, in line 3 'is equal to' means 'is identical to', while in line 5 it means 'can be put in a bijection with'. While in their application to finite multitudes the concepts of equality given through these senses are equivalent, this is not so in the infinite case, for if we substitute the sense of 'is equal to' in line 3 for that of 'is equal to' in line 5, we obtain the falsehood that the multitude of the squares is identical to that of the numbers. Leibniz (who, at times, used the word 'term' instead of 'concept' 13 ) defined Same or coincident terms are those which can be substituted for each other anywhere without affecting truth … Diverse terms are those which are not the same or in which substitution sometimes does not work. [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF])) 14 12. Russell 1919, 80-81. See also Benardete 1964, 47-48, andLevey 1998, 61-62. 13. 'By a term I do not understand the word but the concept or that which the word signifies, you could also say the notion or the idea.' ('Per Terminum non intellego nomen sed conceptum seu id quod nomine significatur, possis et dicere notionem, ideam', [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz , 243 (c. 1680)) )) 14. 'Eadem seu coincidentia sunt quorum alterutrum ubilibet potest substitui alteri salva veritate … Diversa sunt quae non sunt eadem, seu in quibus substitutio aliquando non procedit.' (Leibniz 1875, 7:236) Therefore, by Leibniz's own criterion, the concepts of equality in lines 3 and 5 are diverse or different, as substitution of the one for the other does not preserve truth here; but then, from lines 3 and 5, one cannot infer to line 6, for that inference presupposes that the two concepts are the same. Hence, Leibniz was in a position to see that the inference was incorrect.

Earlier in the text in which he states the proof of the part-whole axiom, Leibniz defines 'Equals are things having the same quantity' [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF](Leibniz , 667 (after 1714)) 15,16 About the concept of quantity, Leibniz there only remarks that it is essentially comparative: 'Quantity or magnitude is that in things which can be known only through their simultaneous compresenceor by their simultaneous perception' (Leibniz 1969, 667). 17,18 However, that leaves unaddressed the fact that there are essentially different ways of comparing, resulting in correspondingly different concepts of equality. 19 Breger has recently attempted to meet this objection to Leibniz's argument by suggesting that Leibniz was working strictly within a theory of finite multitudes: 'The fact that one finds objects outside the theory examined here for which both notions [of equality] are not equivalent is of no importance within the theory' (Breger 2008, 314). However, if that was what Leibniz was doing, then he could not have devised his argument against infinite wholes in the first place, for he then would have had no theory to apply 15. 'Aequalia sunt ejusdem quantitatis.' (Leibniz 1849, 7:19) 16. See also [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6,4:165 (1679) ) and [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6,4:406 (1680(?)(?)). 17. 'Quantitas seu Magnitudo est, quod in rebus sola compraesentia (seu perceptione simultanea) cognosci potest.' [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF](Leibniz , 7:18 (after 1714)) )) 18. See also [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF]Leibniz, -, 6,4:168 (1679)).

19. The example that Leibniz goes on to give in the same text, that of measuring either in inches or in feet, is not a case of two essentially different ways of comparing, because, as Leibniz himself points out (for a different purpose), they are interdefinable. Cantor remarks, after having shown that extending an infinite line (with one endpoint) by a finite line does not lead to a new line with more points, because there will be a 1-1 mapping: Who here, and in the case of any actually infinite quantity, sees a violation of the principle of contradiction, is quite mistaken, by losing sight of the abstractive character of 'magnitude' and wrongly identifying it with the substantial entity of the quantity at hand. 20 Evidently, which abstractions are legitimate in defining a specific concept of magnitude depends on the context: there are perfectly good uses for the concept of magnitude that Cantor in this context rightly rejects. See also my concluding remark. 20. 'Wer hier wie überhaupt bei aktual-unendlichen Quantitäten einen Verstoß gegen das Widerspruchsprinzip findet, irrt durchaus, indem er den abstraktiven Charakter der "Größe" aus dem Auge verliert und sie fälschlich mit der substanziellen Entität des vorliegenden Quantums identifiziert.' [START_REF] Cantor | Mitteilungen zur Lehre vom Transfiniten[END_REF](Cantor [ ] 1932, 393) , 393) to the assumption by which he begins his argument. Leibniz is, on the contrary, working in a theory of parts, wholes and finite as well as infinite multitudes, and within that theory attempts to show that infinite multitudes cannot be wholes. Breger, moreover, holds that the refutation depends on a perspective that was developed only in the second half of the nineteenth century and that for Leibniz 'it would have been absurd, absolutely unthinkable, to reject the equivalence [of the different notions of equality]' (Breger 2008, 315). However, as the above reconstruction emphasises, the refutation uses no concept or technique that was not available to Leibniz. 3 The consistency of Cantorian set theory Arthur, Breger and Brown hold that the refutation of Leibniz's argument depends on whether Cantor's theory of infinite sets is, in fact, consistent:

[The] argument (like those of Cantor, Russell, and Rescher before it) reduces to this: if with Cantor one assumes … [the proposition C] that an infinite collection (such as the set of all numbers) is a whole or unity, then one can establish a consistent theory of infinite number; therefore Leibniz's argument against it is unsound … To say that Leibniz's argument is unsound on the basis of the success of Cantor's theory is to assume the truth [and hence the consistency] of C, and thus to beg the question (unless one has an independent argument for C, which Cantor does not) (Arthur 2001, 105). 21 I suppose that one might argue that, for all we know and he knew, Leibniz's argument against infinite number and wholes might be sound; for despite the fact that most mathematicians now seem to assume that Cantorian set theory is consistent in light of its long record of success and the absence of any proof of its inconsistency, it remains true that neither does there exist a general consistency proof for that theory. It remains at least possible, I suppose, that Leibniz's argument against infinite number and wholes is sound, and that there really is 21. Leibniz (2001, 407n41) adds to this that 'the paradoxes of the infinite still [beset] set theory'. But in the iterative concept of set, which has become the standard understanding of Cantorian set theory, no paradox has yet been found. See Gödel 1947, 518-519 andWang 1974, 181-193. some inconsistency lying dormant and undiscovered in the assumption that the part-whole axiom fails in the case of the infinite-and ultimately some inconsistency lying dormant and undiscovered in the Cantorian definitions of 'less than', 'greater than', and 'equal to' for infinite sets-so that infinite wholes are indeed impossible. (Brown 2005, 486) It is striking that one already needs the existence (free of contradiction) of an infinite totality, which is precisely what is supposed to be proved or refuted, to show the non-equivalence of the two notions of 'having the same number' … The two expressions are equivalent if and only if they are applied to finite multitudes. In other words: one can demonstrate the non-equivalence of the two notions if and only if one assumes the existence of infinite multitudes that as objects free of contradiction constitute a whole and are thus sets (as happens in the Zermelo-Fraenkel theory) or if one has already demonstrated this in some other way. As long as this has not happened, the objection that Leibniz is using two non-equivalent notions is false. (Breger 2008, 313-314) However, it is Leibniz who makes the assumption at the beginning of his argument that an infinite whole exists. Of course, for him that is only an assumption towards a reductio ad absurdum, where the absurdity will arise when the part-whole axiom is brought in. The equivocation on the concept of equality occurs already before that stage of the argument is reached. When Leibniz makes this equivocation, the assumption has not yet been cancelled. Therefore, the need to make that assumption in order to be able to distinguish the two concepts of equality in play is no objection to the refutation of Leibniz's argument. Moreover, as the assumption is part of the argument itself, and not a presupposition of its refutation, the latter does not depend on whether Cantorian set theory in fact is consistent. (This also means that this refutation by itself does not show that there can be no correct arguments against infinite wholes; only that Leibniz's argument is not one.) 4 The part-whole axiom It is often held that the distinction between different concepts of equality on which the refutation of Leibniz's argument depends also serves to show that the existence of infinite wholes and the part-whole axiom are incompatible. 22 Here, too, means available to Leibniz can be indicated that enable one to see that, logically speaking, there is no such incompatibility This turns on the fact that Leibniz defines the concept of proper part in terms of equality: a proper part is a part that is not equal to the whole. The idea now is that, as there are (extensionally) different concepts of equality, concepts whose definition involves the concept of equality may turn out to be equivocal, too. For example, a part of a whole may be proper with respect to one concept of equality and not with respect to another. Likewise, a part may be greater than another one with respect to one concept of equality and not with respect to another. If one accepts this view, then one can argue as follows.

Given that the part-whole axiom relates two concepts (proper part and being greater than) to each other that both involve the equivocal concept of equality, in any particular application of that axiom the same concept of equality should be used throughout. Now consider the axiom in its more explicit form: 'For all x and y, if x is a proper part of y, then y is greater than x'. (Leibniz recognised explicitation of this kind: 'A is B, that is the same as saying that if L is A, it follows that L also is B'; 23 and 'The affirmative universal proposition Every b is c can be reduced to this hypothetical one: If a is b, then a is c'. 24 ) Instantiating the axiom in this form by taking for x the multitude of the squares and for y the whole of the numbers yields the simple conditional 'If the multitude of squares is a proper part of the whole of the numbers, then the whole of the numbers is greater than the multitude of the squares'. Any choice of concept of equality that, via the concept of 'proper part' it induces, makes the antecedent come out true, will, by the definition of 'greater than', make the consequent come out true as well; for example, the concept of equality defined in terms of elementhood. If, on the other hand, the chosen concept of equality makes the antecedent false, such as that defined in terms of a bijection, then it would be open to Leibniz to accept the conditional as vacuously true: the principle that 22. See, for example, the quotation from Brown 2005 in section 2, and Leibniz 2001, 406n41. 23. 'A est B, idem est ac dicere si L est A sequitur quod et L est B.' (Leibniz 1903, 260) 24. 'Propositio Universalis affirmativa Omne b est c reduci potest ad hanc hypotheticam Si a est b, a erit c.' [START_REF] Leibniz | Nouveaux essais sur l'entendement[END_REF](Leibniz, -, 6,4:126 (1678/9(?)/9(?))) conditionals with false antecedents are true can be proved in (a rational reconstruction of) the logical calculus that Leibniz devised in 1690 (Leibniz 1903, 421-423). 25 In neither case does the conditional come out false. This shows that if one defines the concept of proper part in terms of equality, the part-whole axiom and the existence of infinite wholes are not logically incompatible.

Concluding remark

In Cantorian set theory, concepts of size are defined ones, and there is nothing against defining alternatives to Cantor's own, or to working with several at the same time. In general, with different concepts of size will come different principles of arithmetic. (Indeed, Cantor himself has two different concepts of size, cardinality and ordinality, with different arithmetics.) For a mathematical exploration of Cantorian set theory equipped with a concept of size that respects the part-whole axiom non-vacuously for infinite sets, see [START_REF] Benci | An Aristotelian notion of size[END_REF], for an introduction with philosophical and historical background, [START_REF] Mancosu | The Size of Infinite Collections of Natural Numbers: Was Cantor's Theory of Infinite Number Inevitable?[END_REF]

Acknowledgement. I have benefited from discussions of this and related matters with Richard Arthur, Herbert Breger, Leon Horsten, Hidé Ishiguro, Nico Krijn and Robert Tragesser, as well as from the helpful comments of an anonymous BJHP referee.


C H Lamarque

O Janin

MODAL ANALYSIS OF MECHANICAL SYSTEMS WITH IMPACT NON-LINEARITIES: LIMITATIONS TO A MODAL SUPERPOSITION

This paper presents an attempt to generalize the modal superposition formula to mechanical systems with impact type non-linearities following the procedure introduced for smooth non-linearities. The study is restricted to simple one-and two-degree-of-freedom systems with a unilateral constraint on one of the degrees of freedom, for which the response can be analytically determined. Generalized frequencies, modes and masses are built in the procedure. The results obtained for various sets of parameters point out some limitations to the validity of a general modal superposition formula.

INTRODUCTION

Analysis of the response of structures is convenient if a linear model can fully describe the structure. Within this framework, it is useful to introduce in either the "nite or in"nite dimension (Hilbertian case) the notion of eigenmodes of the structure. They are either normal modes (de"ned by adding conservative conditions to the model) or complex modes (taking into account viscous damping for example) [1,[START_REF] Meirovitch | Analytical Methods in <ibrations[END_REF]. The linear theory of di!erential systems provides the response of the structure to an external elementary sinusomK dal excitation in an interesting form: the full response is simply the superposition of the responses of each mode to the excitation. Such a formula is well known; this is the superposition formula which is the basis of modal synthesis [START_REF] Meirovitch | Analytical Methods in <ibrations[END_REF][START_REF] Jezequel | theH orie et extensions[END_REF]. The notion of modal synthesis can be extended to the case of sub-structures by using linear operator theory [START_REF] Jezequel | theH orie et extensions[END_REF][START_REF] Bourquin ; Ph | Synthe`se modale et analyse numeH rique des multi-structures eH lastiques[END_REF].

In the non-linear case, the notion of non-linear modes had been considered "rst. In the case of mathematically smooth non-linearities and for a "nite number of degrees of freedom (d.o.f.) with particular polynomial non-linearities, Rosenberg and others "rst introduced natural modes [START_REF] Rosenberg | [END_REF] and then non-linear normal modes [6}9], and investigated their stability. Since then, many methods have been used to introduce modes (natural, non-linear, non-linear normal, minimal normal, non-linear similar normal, etc.) in the case of non-linear structures; methods derived from the works of Rosenberg [10}13]; a stroboscopic method [14]; methods based on averaging and modal truncation [15}17]; direct or geometrical methods for conservative systems [18}20] or in the Hamiltonian frame [21}25]; PadeH approximation [26,[START_REF] Vakakis | Normal Modes and ¸ocalization in Nonlinear Systems[END_REF]; multi-spectral, Volterra series and HFRFs [START_REF] Chen | New Advances in Modal Synthesis of ¸arge Structures[END_REF][START_REF] Gifford | [END_REF]; integral transforms [30]; Lie series [31]; methods based on normal forms in the Hamiltonian [32}34] or general frame [35}40]; or methods using centre manifold theory [41}47] or amplitude equations [48]. JeH zeH quel and Lamarque extended the modal superposition starting from non-linear modes built via normal forms, for systems with a few d.o.f. and smooth non-linearities [49}51], even for the case of complex modes [START_REF] Lamarque | Proceedings of the Conference on Experimental and Numerical Methods in Structural Dynamics[END_REF]; this method is valid for su$ciently small non-linear oscillations. The non-linear modes and generalized masses obtained depend on the amplitudes of the normal co-ordinates. They are built up in order to agree at best with a resonance equation. The modes do not always verify the reciprocity condition which exists in the linear case.

The question of non-linear modes is obviously related to the search for periodic solutions to non-linear dynamical systems and the study of their stability [START_REF] Lamarque | Proceedings of the Conference on Experimental and Numerical Methods in Structural Dynamics[END_REF][START_REF] Rand | [END_REF]. It therefore involves a huge amount of literature dealing with numerous analytical methods, perturbation methods, and methods for bifurcation analysis for example references [32, 54}66].

In the "eld of stochastic behaviour, the question of non-linear modes has been examined already [START_REF] Sri | [END_REF] and tools are available [START_REF] Boxler | How to Construct Stochastic Center Manifolds on the level of <ector Fields[END_REF]. In the case of non-smooth non-linearities, only particular cases have been investigated; that is, normal modes for piecewise linear systems [START_REF] Chen | Normal modes for piecewise linear vibratory systems[END_REF][START_REF] Zuo | Proceedings of Euromech[END_REF]. Sometimes, in order to deal with localized or weak non-linear non-smooth phenomena using linear methods, one can introduce a modi"ed dissipation or sti!ness matrix.

From the point of view of dynamics, vibro-impact systems have been thoroughly studied in the literature: global behaviours and periodic solutions have been investigated in the single-d.o.f. case (both analytically and numerically in references [71}74] or by means of a change of variables in reference [75]) and in the two-d.o.f. case (double impact oscillator in references [START_REF] Peterka | Proceedings of the Second European Nonlinear Oscillation Conference[END_REF][START_REF] Peterka | Proceedings of the 1999 ASME Design ¹echnical Conferences[END_REF] or impact damper in references [START_REF] Chatterjee | Bifurcations and chaos in autonomous self excited oscillators with impact damping[END_REF][START_REF] Sung | [END_REF]). Singularities in the dynamics of such systems have been pointed out in references in [80}82], and some authors have examined the e!ect of dry friction on mechanical systems with impacts in [83}86]. General results for vibro-impact systems can also be found in references [START_REF] Brogliato | Nonsmooth Impact Mechanics[END_REF][START_REF] Babitsky | ¹heory of <ibro-Impact Systems and Applications[END_REF]. Moreover, a modal approach has been introduced in reference [START_REF] Mikhlin | [END_REF] to deal with direct and inverse problems in discrete systems with impacts, based on the theory of non-linear normal modes (see reference [START_REF] Vakakis | Normal Modes and ¸ocalization in Nonlinear Systems[END_REF]).

Nevertheless, to our knowledge, no attempt to build a modal superposition similar to the linear case exists in the case of hard non-smooth non-linearities such as friction or impact. The main aim of this work is to seek some answers to the question about possibility of building a modal superposition and a modal synthesis in the case of structures exhibiting a non-linearity of impact type. The problem is considered for the case of simple systems with one and two d.o.f.

In section 2, a single d.o.f. system is considered. Using a piecewise exact integration (2.1), the periodic responses under sinusomK dal excitation are studied (2.2). The building of a modal superposition in equation (2.3) is then tested by introducing successively a generalized eigenfrequency from free vibrations, a generalized mode and a generalized mass associated with forced oscillations.

In section 3, two-d.o.f. systems are considered. First the case of weak coupling and impact of a mass against a rigid stop is dealt with (3.1). Then the case of strong coupling is examined, again with impacts against a external rigid stop (3.2). The results obtained are applied to the case of direct impacts between two rigid solids (3.3). Finally, in section 4 conclusions are drawn on the relevance of the modal superposition formula obtained.

SINGLE-DEGREE-OF-FREEDOM SYSTEM

The system studied consists of a single-d.o.f. damped harmonic oscillator with a unilateral constraint, for which an impact law is de"ned (see Figure 1). The impact process is considered to be instantaneous and the behaviour of the system at the time of impact is described using the coe$cient of restitution e3[0,1], characterizing the energy loss during impact. The equations governing the dynamics of the system are then

xK #ax # x"f cos ( t), x (t))x K?V , (1)
x (t>)"!exR (t\) if x (t)"x K?V .

If the velocity immediately before the impact at tN is zero, several cases can occur: if at tN the acceleration is negative, then the system is still described by equation ( 1) after the impact and the trajectory is tangent to the stop at tN . If, on the other hand, the acceleration is positive, then the system remains in contact with the stop for a non-zero time interval. It can be shown that sticking never occurs if f( x K?V . In the following, it is assumed that the system's parameters satisfy this condition.

ANALYTICAL SOLUTION

The system with sinusomK dal forcing can be written:

xK #ax # x"f cos ( t), x (t>)"!ex (t\) if x (t)"x K?V , (2)
x (0)"x , xR (0)"x .

Since this system is linear between two consecutive impacts, it is possible to determine a piecewise analytical form of the solution on 1>. Setting "( !a/4 and "a/2 . ∀k3-*, the solution on [t I\ , t I ] can be written in the form

x(t)"e\?R [A I cos ( J t)#B I sin ( J t)]#f cos ( t)#f sin ( t), x(t> I )"x (t\ I )"x K?V , (3)
x (t>

I )"!ex (t\ I ),
where t "0 and

f "f ! ( ! )#a , f "f a ( ! )#a .
From equations (3), the following recursive relation gives the values of the constants A I and B I :

A B " x !f x !f # (x !f ) , A I> B I> " A I B I #(1#e) u (A I , B I , t I ) !sin ( t I ) cos ( t I ) , (4)
u(A I , B I , t I )"[sin ( J t I )# cos( J t I )] A I #[!cos ( J t I )# sin ( J t I )] B I # J e at I /2 [ f sin ( t I )!f cos ( t I )].

PERIODIC SOLUTIONS

Owing to the analytical form of the solution given by equations ( 3) and (4) it is possible to seek analytically a periodic solution, similar to that achieved in reference [72] or [START_REF] Peterka | [END_REF] for example. A solution of period n¹ with k impacts per cycle is here called (n, k)-periodic, where ¹"2 / is the period of the external excitation.

(n, 0)-periodic solutions

The simplest case consists of looking for n¹-periodic solutions which never impact against the stop. Such a case implies n"1 and the initial conditions leading to a (1,0)-periodic solution are and as it is assumed that f( x K?V , it can be shown that (n, 0)-periodic solutions exist if and only if ) \ or * > . The stability of these periodic solutions can be determined using a PoincareH map de"ned by a constant phase plane Z"¹ in the co-ordinates (X, >, Z)"(x, x , t mod ¹). As pointed out in reference [80], such a mapping is not everywhere continuous nor di!erentiable. Therefore, for each periodic solution corresponding to a "xed point of the PoincareH map where it is continuously di!erentiable, the stability can be investigated. It can be shown that (1, 0)-periodic solutions are always stable.

x "f , x "f . Setting: > " J ! a 4 # f x K?V !a J , \ " J ! a 4 ! f x K?V !a J (5)

(n, 1)-periodic solutions

Similar to section 2.2.1, n¹-periodic solutions with one impact per cycle can be sought by using equations ( 3) and (4). In this case, the impact time can be analytically determined, and the initial conditions leading to (n, 1)-periodic solutions are then given by

x "A (t )#f , x " [B (t )! A (t )]# f ,
where A (t ) and B (t ) depend analytically on the system's parameters. An example of (1, 1)-periodic solution is shown in Figure 2.

As in section 2.2.1, the PoincareH map can be used (when it is de"ned and of class C locally) in order to determine the type of the periodic solutions obtained (see in Figure 3). The Jacobian matrix can be calculated analytically from the analytical form of the PoincareH map by taking into account the in#uence of the partial derivatives of the impact time t with respect to the initial conditions.

It can be shown that such a stability study is valid only if O > and O \ for in that case the PoincareH map is not di!erentiable at the "xed point considered.

(n, 2)-periodic solutions

The method for seeking n¹-periodic solutions with two impacts per cycle is identical to the case with one impact per cycle, with a new unknown due to the second impact time t . By characterizing the periodicity of the solution and by using the analytical form (3), (4) a system of two non-linear equations with two unknowns (t , t ) can be obtained which can for example be solved using Newton's method. Once the value of t and t are known, the initial conditions of the system are given by

x "A (t , t )#f , xR " J (B (t , t )! A (t , t
))# f .

Figures 4}6

show three examples of (n, 2)-periodic solutions that can be obtained analytically.

As in the case of (n, 1)-periodic solutions, the stability of the (n, 2)-periodic solutions can be studied by using the PoincareH map: when , + > , \ ,, it is possible to calculate analytically the Jacobian matrix of the PoincareH map, and its eigenvalues determine the stability or instability of the periodic solution.

MODAL SUPERPOSITION

Free oscillations

The free oscillations of the system are described by the following equation:

xK #axR #
x"0, xR (t>)"!exR (t\) if x (t)"x K?V , (6)

x (0)"x , xR (0)"xR .

The solution of this equation can also be piecewise analytically written: the expression is identical to equation (3), the constants of integration being given by equation ( 4), with f "f "0. If the equilibrium is a position that the system can physically reach, namely if x K?V '0, and if aO0, it can be shown that the system (6) has a "nite number K of impacts. This result is interesting from the modal point of view; it means that the free oscillations of the system are governed by the frequency , except for a bounded time span. Hence, it will be considered later on that the natural frequency of the system is . Remark. When a"0, the number of impacts is in"nite and lim

I> t I> !t I "2 / J . If x K?V "0
, there also is an in"nite number of impacts and t

I> !t I " / J , ∀k. Finally, if x K?V (0 then lim I> t I> !t I "0.

Generalized mass and modal superposition

In this section, it is intended to establish a modal superposition formula for the irregular non-linear model previously introduced. The case x K?V '0 is considered for which the

natural frequency is

, and an attempt is made to obtain a modal superposition formula similar to the linear case, connecting the nth harmonic amplitude of the response to the amplitude of the forcing.

If a (n, k)-periodic solution is considered, i.e. a solution with period n¹ and k impacts per cycle. Due to the n¹-periodicity of the response, the Fourier coe$cients can be de"ned; for all j39 we have

c H ( )" 1 n¹ L2 x (t) exp !i j n t dt " 1 n¹ t x (t) exp !i j n t dt# I\ K t K> t K x (t) exp !i j n t dt # L2 t I x (t) exp !i j n t dt .
x (t) is known to be piecewise via equation [START_REF] Jezequel | theH orie et extensions[END_REF]. These k#1 integrals can then be calculated analytically. If kO0 de"ne

HH LI ( )" 2 J n¹ +B #A I> e\?L2 sin (n J ¹)!B I> e\?L2 cos (n ¹) #( #i H ) [A !A I> e\?L2 cos (n J ¹)!B I> e\?L2 sin (n ¹)] (7)
#(1#e) I K e !at K /2 exp !i j n t K u (A K , B K , t K ) ,
where H "j /n J , and if k"0, HH L ( )"0. Then

c H ( )" f !if 2 L H # f !if 2 \L H # HH LI ( ) 2 . ( 8
) Let I ( )" ! #ai . ( 9
)
The nth Fourier coe$cient is then given by

c L ( )" f !if 2 # HL LI ( ) 2 I ( ) " f#HL LI ( ) 2 I ( ) . ( 10
)
Let the modal mass m LI be:

m LI " 1 1#HL LI ( )/f . ( 11
)
Because the free damped steady state oscillations are linear oscillations, the generalized mode is here represented by the scalar 1. The nth harmonic amplitude can then be written in the form

A L ( )"(2 ("c L "#"c \L ")"2"c L "" f m LI I ( ) . ( 12
)
This is a modal superposition formula connecting the nth harmonic amplitude of the forced response to the amplitude of the forcing via the free response. This formula is similar to the formula that obtained in the linear case, but the mass (which should be equal to 1) is replaced by a modal mass (11).

Due to this de"nition, the mass is a complex: to give it a more physical meaning, it is necessary to consider its module. Thus, the unitary mass system with impact is modelled and, subjected to a sinusomK dal forcing of frequency , like a linear system without impact of mass "m LI ", subjected to the same forcing. This modelling holds in terms of spectral amplitude for a (n, k)-periodic response: the spectral amplitude is the same one for both systems.

Remark.

* In the case of a n¹-periodic response without any impacts, it was seen that only the case n"1 was possible. Thus, m "1 which is coherent since in this case the classical modal superposition formula applies and gives A ( )"" f/ I ( )".

* There is no longer a unique modal superposition formula as in the linear case, but an in"nity a priori; it depends on the period and number of impacts per cycle of the periodic solution.

* Preceding calculations give access to the whole Fourier series of a (n, k)-periodic response.

* An analytical expression of the module of the modal mass is obtained if an only if k"1, i.e., when the periodic solution has only one impact by period. Setting

X " 2 (1#e) nf ¹ !1#2e?L2 cos (n ¹)!e?L2 e#e?L2 +(1!e) cos (n J ¹)#(1#e) sin(n ¹),!e?L2 ;[ f sin( t )!f cos ( t )]
for the modal mass:

m L " 1 1#X e !i t ( 13
)
Since t is analytically known in the case of a (n, 1)-periodic response, the modal mass "m L " can be expressed analytically as a function of the parameters of the system.

Examples of modal superposition

It is now possible to show some applications of the modal superposition formula previously established, using the analytical search for periodic solutions carried out in section 2.2. First consider the following set of parameters for the system:

"2)5, a"0)05, x K?V "14, e"0)9 and f"20. Note that for these values of parameters x K?V 'f and e'0, so that sticking never occurs.

Figures 10 outlines an important di!erence compared to the linear case: the peak of amplitude corresponding to resonance does no longer exist around . Consequently, for close to the natural frequency of the system, the modal mass is the largest because there is no resonance between the external forcing and the system. The maximum amplitude occurs at a frequency higher than the natural frequency, but the maximum reached is much weaker than in the linear case; there is not a true peak of amplitude in the usual sense.

In addition, the modal superposition formula previously established enables the amplitude of nth harmonic of a solution, whose period is n¹ to be computed. In the linear case (without impact), this is su$cient to know the whole spectral response of the system: only n"1 occurs and the Fourier coe$cients are all zero except the "rst one. In this case, it is no longer su$cient, for the occurrence of impacts leads to periodic solutions with many harmonics, and the nth harmonic amplitude can be a poor approximation to the spectral amplitude of the response. For example, Figure 12 shows that the second-harmonic amplitude for a ¹-periodic solution can be the largest one. In the same way, the Fourier coe$cient c can become large, can be seen in Figure 10. Table 1 summarizes the di!erence between the spectral amplitude and the nth harmonic amplitude for various periodic solutions.

For this example, a modal superposition formula can be built and is worthwhile as long as remains close to the &&primary resonance''. Nevertheless, this frequency area does not always correspond to the largest amplitudes. An example showing a spectral behaviour di!erent from the preceding one can now be considered. In this case "1, a"0)02, x K?V "1, e"0)9 and f"20 (sticking to the stop may then occur, but only responses without sticking will be considered).

In Figure 18 two important characteristics can be seen. First of all, and contrary to the previous example, the response exhibits a true peak of spectral amplitude similar to the resonance peaks observed in linear systems. However, the peak does not occur at the natural frequency , but at +2 . Furthermore, the approximation to the whole amplitude by the amplitude of "rst harmonic is very rough: resonance occurs through the Fourier coe$cient c of the solution, and the remainder of the harmonics are negligible in the neighbourhood of the peak. The last two examples have shown that the building of a modal superposition formula similar to the linear case comes up against two major di$culties. On one hand, the multiplicity of types of periodic solutions prevents a unique formula which holds for any case: it is necessary to know a priori the period of the response and the number of impacts per cycle. On the other hand, taking into account only one harmonic in this superposition may not be su$cient any more; it would be necessary to include at least the four "rst harmonics in the formula in order to get a closer approximation in the examples shown.

TWO-DEGREE-OF-FREEDOM SYSTEMS

This section deals with a system with two d.o.f., one of them being constrained by a stop:

xK #axR # x#k y"f cos ( t), x3C (1>,]!R, x K?V ]), x (t)"x K?V NxR (t>)"!exR (t\), yK #ayR # y#k x"f cos ( t), y3C(1>, 1).
This system can be written in the form

X G #AX #KX"F cos ( t)#impact, (14)
with

X" x y , A" a 0 0 a , K" k k , F" f f
and the term &&# impact'' represents the constraint induced on x by the occurrence of impacts.

The model just introduced is a general mathematical model. It will be studied in three stages: "rst of all, consider the case k "0 (which will be referred to as &&weak coupling'') for which the study is very close to the single-degree-of-freedom case in term of periodic solutions: it will give an initial outline of modal superposition with two d.o.f.

The general case where k O0 (referred to as &&strong coupling'') will then be studied, for which the modal superposition is similar to the case k "0, except that the search for periodic solutions becomes more complex. Finally it can be seen how the modal superposition can be written in the case of two rigid bodies colliding.

Note, initially, that the "rst case is a mathematical model which cannot easily be expressed in mechanical terms. Indeed, the action}reaction principle prevents x from acting on y without y acting on x. Therefore, it is necessary to consider carefully the physical conclusions that could be drawn from this model.

WEAK COUPLING

System (14), the special case where k "0, which is close to the single-d.o.f. case previously studied can be dealt with. and pre-multiply the system 14 without impacts by P\, a system in the new co-ordinates x x "P\X is obtained:

xK #axR # x "f cos ( t), xK #axR # x "f cos ( t), ( 15
)
where

f "f , f "!v f #f .
Written in that form, the system is easy to solve and as long as there is no impact:

x (t)"e\?R [A cos ( J t)#B sin( J t)]#f cos ( t)#f sin ( t), x (t)"e\?R [A cos ( J t)#B sin ( J t)]#f cos ( t)#f sin ( t), (16)
where "( !a/4, J "( !a/4 and

f "f ! ( ! )#a , f "f a ( ! )#a , f "f ! ( ! )#a , f "f a ( ! )#a .
Moreover, if "a/2 J and "a/2 J , the solution is given in the initial co-ordinate system by

x"x ,

y"v x #x . (17)
3.1.1.2. Gluing at impact times. An impact occurs when x (t)"x K?V , which is equivalent to x (t)"x K?V . If it is assumed that there is an impact at t I , on [t I\ , t I ], the solution ( 16) is given by

x (t)"e\?R [A I cos ( J t)#B I sin ( J t)]#f cos ( t)#f sin ( t), x (t)"e\?R [A I cos ( J t)#B I sin ( J t)]#f cos ( t)#f sin ( t).
From this equation it can be inferred that t I is solution of the equation

f (t I )"e !at I /2 [A I cos ( J t I )#B I sin ( J t I )]#f cos ( t I )#f sin ( t I )!x K?V "0.
Moreover, by assumptions on x and y, at impact time

x(t> I )"x(t\ I ), xR (t> I )"!ex (t\ I ), y (t> I )"y (t\ I ), yR (t> I )"yR (t\ I ),
which yields for the new variables:

x (t> I )"x (t\ I ), xR (t> I )"!exR (t\ I ), x (t> I )"x (t\ I ), xR (t> I )"xR (t\ I )#(1#e) v x (t\ I ).
Thus, obtain the following relationships provide the constants of integration

A I> "A I !(1#e) sin ( J t I ) u (A I , B I , t I ), B I> "B I #(1#e) cos ( J t I ) u(A I , B I , t I ), A I> "A I #(1#e) v J J sin ( J t I ) u (A I , B I , t I ), (18)
B I> "B I !(1#e) v J J cos ( J t I ) u(A I , B I , t I ),
where u is given by

u (A, B, t)"A [sin( J t)# cos( J t)]#B[!cos ( J t)# sin ( J t)] #e?R J [f sin ( t)!f cos ( t)].

Search for periodic solutions

The search for periodic solutions (x, y) is equivalent to the search for periodic solutions (x , x ). According to equation ( 17), x "x, therefore the search for periodic solutions for x is similar to the one carried out in the case of a single-d.o.f. system. Thus, it is possible to determine x and x so that x is (n, k)-periodic where k3+0, 1, 2,. As regards x , the method is identical; only the recursion that gives A and B being di!erent. Thus 2;2 linear systems are obtained in (A (t ), B (t )). When the determinant is non-zero, x and xR are obtained so that x is (n, k)-periodic where k3+0, 1, 2,. The initial conditions for the original system co-ordinates (x, y) are given by equation (17):

x "x , y "v x #x , xR "xR , yR "v xR #xR . 3.1.3. Modal superposition 3.1.3.1.
Free oscillations of the system. It was seen in Section 2.3.1 that, in the case of a single-d.o.f. system without external forcing, the number of impacts is "nite and the steady state response is periodic with frequency . Hence, in the case of weak coupling, the free response also exhibits a "nite number of impacts, and the steady state response for x is periodic with frequency . As for x , since there are no more impacts once the steady state response is reached, the equation of its movement is given by equation (15):

xK #ax # x "0.
This is the equation of a classical damped oscillator without external forcing, whose steady state response has frequency . In order to establish a modal superposition formula, it is necessary to start from the natural frequencies and . Moreover, in that case again, the generalized modes correspond to linear modes.

3.1.3.2. Generalized masses and modal superposition. Consider a (n, k)-periodic solution. The Fourier coe$cients of the functions x and x can be calculated. These are quite similar to the coe$cients found for the single-d.o.f. system. Setting

HH LI ( )" 2 J n¹ 1 ! j n #ia j n B #A I> e\?L2 sin (n J ¹)!B I> e\?L2 cos (n J ¹) #( #i H ) [A !A I> e\?L2 cos (n J ¹)!B I> e\?L2 sin (n J ¹)] #(1#e) I K e !at K /2 exp !i j n t K u (A K , B K , t K ) GH LI ( )" 2 J n¹ 1 ! j n #ia j n ; B #A I> e\?L2 sin (n J ¹)!B I> e\?L2 cos (n J ¹) #( #i H ) [A !A I> e\?L2 cos (n J ¹)!B I> e\?L2 sin (n J ¹)] !(1#e) v J J I K e !at K /2 exp !i j n t K u (A K , B K , t K )
with H "j /n J and H "j /n J , the Fourier coe$cients are given by

c H ( )" f !i f 2 L H # f #if 2 \L H # HH LI ( ) 2 , c H ( )" f !if 2 L H # f #i f 2 \L H # GH LI ( ) 2 .
The Fourier coe$cient corresponding to the nth harmonic are inferred by using the coordinate transformation (17

) cV L ( )"c L ( )" f 2mLI I , cW L ( )"v c L ( )#c L ( )"v f 2mLI I # f 2mLI I , ( 19
)
where, I " ! #ai , I " ! #ai and

mLI " 1 1#HL LI ( )/f , mLI " 1 1#GL LI ( )/f . ( 20
)
The nth Fourier coe$cients are then in the original basis:

cV L ( )" f 2mLI I , cW L ( )"v f 2mLI I # !v f #f 2mLI I .
The contribution of the nth harmonic to the Fourier spectrum is given (as seen in equation ( 12)) by

A6 L ( )""X L ( )"" f mLI I , A7 L ( )""> L ( )"" v f mLI I # !v f #f mLI I
, (21) which can also be written as

X L ( ) > L ( ) " 1 0 v 0 mLI I f f # 0 0 !v 1 mLI I f f . ( 22
) If A " 1 0 v 0 and A " 0 0 !v 1
"0, a a "1, b a "v .
It is thus possible to arbitrarily set one of the unknowns; for example, as in the case of linear modes, if a

"1, then a "1, b "v , hence T "(1, v ), T "(1, 0). ( 23
)
The same type of calculations and assumptions for A lead to T "(0, 1),

T "(!v , 1). ( 24
)
Using equations ( 23) and (24), relation (22) can then be written in the form

X L ( ) > L ( ) " TR T mLI I ( ) f f # TR T mLI I ( ) f f . ( 25
)
This has just established a modal superposition formula holding for a two-d.o.f. system with weak coupling. This formula links the nth harmonic amplitude to the free response and the forcing by means of a modal mass. The vectors T , T , T and T represent the modes (left and right respectively). Note that T and T are merely the eigenvectors associated with and . Finally, as in the case of the single-d.o.f. system, there is no unique modal superposition formula which holds in all cases, but according to the period of the response and the number of impacts per cycle, it is necessary to choose the right formula.

Remark. The modal superposition formula seems to exhibit a reciprocity breaking compared to the linear case: ¹ and ¹ are di!erent whatever choice is made when setting one of the unknowns. Indeed, the second component of ¹ is always zero whereas that of ¹ is always non-zero. This is in fact due to the choice of the model, for which k "0 and k O0.

3.1.3.3.

Examples of modal superposition. For the "rst example, the following parameters are chosen: "2)5, "3)8, a"0)05, e"0)9, x K?V "14, f "20, f "18 and k "1. The second example is derived from the single-d.o.f. case and the chosen parameters are the following: "1, "3)8, a"0)02, e"0)9, x K?V "1, f "20, f "18 and k "1. The spectral amplitude for x is the same one as that obtained in the single-d.o.f. case. The continuous d.o.f. y exhibits a resonance for K similar to the linear case, but several additional secondary resonances occur for K /2 and K /3. Furthermore, for the frequency associated with a spectral amplitude peak for x, a small peak for y is found, which can become large if k is su$ciently large. Lastly, if is close to /2 and k is rather large, then the main peak of resonance can occur in the neighbourhood of /2.

STRONG COUPLING

In this section, the system ( 14) with k O0 is studied.

3.2.1.

Analytical solution of the system 3.2.1.1. Decoupling equations. System (14) will be decoupled in order to be able to write the solutions x and y explicitly. The matrix K has the following characteristic polynomial: The system parameters will be chosen so that '0. Then the matrix K admits two distinct real eigenvalues:

P ) ( )" !( # ) # !k k with discriminant "( ! )#4k k .
" # !( 2 , " # #( 2 .

Assume that

O ; if v "k /( ! ) and v "k /( ! )
, then two eigenvectors associated with and are respectively ( v ) and ( v \ ), which de"ne the matrix

P" 1 !v v 1 .
The assumption '0 implies that its determinant is nonzero, and P is invertible with

P\" 1 1#v v 1 v !v 1 .
Then, if X"PY, the system ( 14) is multiplied by P\ in order to obtain

Y G #aY #P\KPY"P\F cos ( t)#impact Let f " f #v f 1#v v , f " f !v f 1#v v .
The new system obtained using the new basis is given by

xK #axR # x "f cos ( t), xK #axR # x "f cos ( t) (26)
as long as there is no impact. Thus these equations can be solved easily;

x (t)"e\?R [A cos ( J t)#B sin ( J t)]#f cos ( t)#f sin ( t), x (t)"e\?R [A cos ( J t)#B sin ( J t)]#f cos ( t)#f sin ( t) (27)
with J "( !a/4, J "( !a/4 and

f "f ! ( ! )#a , f "f a ( ! )#a , f "f ! ( ! )#a , f "f a ( ! )#a .
The solution in the original basis is "nally given by

x"x !v x , y"v x #x . (28)
3.2.1.2. Gluing solutions at impact times. An impact occurs when x(t)"x K?V , namely when x (t)!v x (t)"x K?V . Assume that there is an impact at t I . For t3[t I\ , t I ], the decoupled solution is given by

x (t)"e\?R [A I cos ( J t)#B I sin ( J t)]#f cos ( t)#f sin ( t), x (t)"e\?R [A I cos ( J t)#B I sin ( J t)]#f cos ( t)#f sin ( t).
From these equations it can be inferred that the equation veri"ed by t I is

f (t I )"e !at I /2 +A I cos ( J t I )#B I sin ( J t I )!v [A I cos ( J t I )#B I sin ( J t I )], #( f !v f ) cos ( t I )#( f !v f ) sin ( t I )!x K?V "0.
Yet by assumptions on x and y, at the impact time:

x (t> I )"x (t\ I ), x (t> I )"!exR (t\ I ), y(t> I )"y(t\ I ), y (t> I )"y (t\ I )
which yields, according to (28),

x (t> I )"x (t\ I ), xR (t> I )" 1! 1#e 1#v v x (t\ I )# (1#e)v 1#v v xR (t\ I ), x (t> I )"x (t\ I ), x (t> I )" (1#e)v 1#v v xR (t\ I )# 1#e 1#v v !e xR (t\ I ).
For i3+1, 2,,

u G (k)"AG I sin ( J G t I )# G cos ( J G t I )]#BG I [!cos ( J G t I )# G sin ( J G t I )] # J G e at I /2 [ f G sin ( t I )!f G cos ( t I )],
then:

A I> "A I ! 1#e 1#v v sin ( J t I ) u (k)!v J J u (k) , B I> "B I # 1#e 1#v v cos ( J t I ) u (k)!v J J u (k) , A I> "A I # (1#e) v 1#v v sin ( J t I ) J J u (k)!v u (k) , (29)
B I> "B I ! (1#e)v 1#v v cos ( J t I ) J J u (k)!v u (k) .

Search for periodic solutions

The search for periodic solutions (x, y) is equivalent to the search for periodic solutions (x , x ). Only (n, 0) and (n, 1)-periodic solutions will be sought. The following calculations are similar to those found in reference [START_REF] Peterka | Proceedings of the Second European Nonlinear Oscillation Conference[END_REF], with the addition of damping.

3.2.2.1. (n, 0)-periodic solutions. As for the single d.o.f system, it is easy to prove that the system admits a (n, 0)-periodic solution if, and only if, x and x are given by

x (t)"f cos ( t)#f sin ( t), x (t)"f cos ( t)#f sin ( t).
According to equation ( 28):

x(t)"( f !v f ) cos ( t)#( f !v f ) sin ( t).
It is thus possible to get results similar to those obtained for a single-d.o.f. system in section 2.2.1. This gives a condition that a (n, 0)-periodic solution exists; it requires

( f !v f )#( f !v f ))x K?V .
From this inequality, a fourth degree polynomial in can be obtained, allowing the values of for which the system admits a (n, 0)-periodic response to be determined. The remark made in Section 2.2.1 still holds; the existence of (n, 0)-periodic solutions requires x K?V '0.

3.2.2.2. (n, 1)-periodic solutions. n¹-periodicity requires

x (n¹)"x , xR (n¹)"xR , x (n¹)"x , xR (n¹)
"xR or using linear combination, for a solution with one impact per cycle:

A cos (n J ¹)#B sin (n J ¹)!e?L2 A "0, A cos (n J ¹)#B sin (n J ¹)!e?L2 A "0, A sin (n J ¹)!B cos(n J ¹)#e?L2B "0, (30)
A sin (n J ¹)#B cos (n J ¹)!e?L2 B "0.
Using equation ( 29

f (t )"e !at /2 +A (t ) cos ( J t )#B (t ) sin ( J t ) !v [A (t ) cos ( J t )#B (t ) sin ( J t )], #( f !v f ) cos ( t )#( f !v f ) sin ( t )!x K?V "0.
Initial conditions of the system leading to a (n, 1)-periodic solution are then given by

x "A (t )#f !v (A (t )#f ), y "v (A (t )#f )#A (t )#f , xR " J [B (t )! A (t )]#f !v J [B (t )! A (t )]!v f , y "v J [B (t )! A (t )]#v f # J [B (t )! A (t )]#f . 3.2.3. Modal superpositon 3.2.3.1.
Free oscillation of the system. It can be shown that the system (14) without external forcing has a "nite number of impacts. Thus the steady state of the system (26) with f "f "0 is periodic with frequency for x and for x : these two frequencies will be used as natural frequencies for the forced system.

HH LI ( )" 2 J n¹ 1 ! j n #ia j n ; B #A I> e\?L2 sin (n J ¹)!B I> e\?L2 cos (n J ¹) #( #i H ) [A !A I> e\?L2 cos (n J ¹)!B I> e\?L2 sin (n J ¹)] # 1#e 1#v v I K e !at K /2 exp !i j n t K u (m)!v J J u (m) GH LI ( )" 2 J n¹ 1 ! j n #ia j n B #A I> e\?L2 sin (n J ¹)!B I> e\?L2 cos (n J ¹) #( #i H ) [A !A I> e\?L2 cos (n J ¹)!B I> e\?L2 sin (n J ¹)] ! (1#e) v 1#v v I K e !at K /2 exp !i j n t K J J u (m)!v u (m) with H "j /n J and H "j /n J . Then c H ( )" f !if 2 L H # f #i f 2 \L H # HH LI ( ) 2 , c H ( )" f !i f 2 L H # f #i f 2 \L H # GH LI ( ) 2 .
The nth Fourier coe$cients of the system by the basis change [START_REF] Chen | New Advances in Modal Synthesis of ¸arge Structures[END_REF] are then

cV L ( )"c L ( )!v c L ( )" f 2mLI I !v f 2mLI I , cW L ( )"v c L ( )#c L ( )"v f 2mLI I # f 2mLI I , (31)
where I " ! #ai , I " ! #ai and

mLI " 1 1#HL LI ( )/f , mLI " 1 1#GL LI ( )/f . ( 32
)
In the original basis,

cV L ( )" 1 1#v v f #v f 2mLI I !v f !v f 2mLI I , cW L ( )" 1 1#v v v f #v f 2mLI I # f !v f 2mLI I .
The contribution of the nth harmonic is (as for the weak coupling in equation ( 21)) given by Three equations with four unknowns are obtained

X L ( ) > L ( ) " 1 1#v v 1 v v v v mLI I f f # v v !v !v 1 mLI I f f . ( 33
) Let A " 1 1#v v 1 v v v v and A " 1 1#v v v v !v !v 1
a a " 1 1#v v , a b " v 1#v v , b b " v v 1#v v , b a " v 1#v v .
As in the case of weak coupling, it is possible to "x one of the unknows; for example, in order to respect a reciprocity condition as long as possible, let a "1/((1#v v ). Then 1#v v '0, for the parameters of the system verify (

! )#4k k '0. Then a " 1
(1#v v , b " v (1#v v , b " v (1#v v . and T " 1
(1#v v (1, v ), T " 1
(1#v v (1, v ). (34)
In the same way, for A :

T " 1 (1#v v (!v , 1), T " 1
(1#v v (!v , 1). (35)
Using equations ( 34) and ( 35), the relation (33) then becomes

X L ( ) > L ( ) " ¹R ¹ mLI I f f # ¹R ¹ mLI I f f . ( 36
)
Thus a modal superposition formula for the two-d.o.f. system with strong coupling has been established.

Remark.

* ¹ and ¹ thus de"ned are in fact eigenvectors associated with and .

* ¹ and ¹ are orthogonal, just as ¹ and ¹ .

* There is no reciprocity when k Ok , but when the matrix K is symmetrical, reciprocity occurs since then ¹ "¹ and ¹ "¹ .

TWO RIGID BODIES COLLIDING

In this section, a mechanical system consisting of two oscillating rigid bodies which can collide during their movement is studied.

It will be assumed that the equilibrium position x and x of the two solids are such that x !x "x K?V '0 (system without preload). The equations of the system in relative displacements are then

m xK #c x # x "g cos ( t), m xK #c x # x "g cos ( t), x !x *x K?V . Let a "c /m , " /
x !x *x K?V .

When x (t)!x (t)"x K?V , an impact occurs at t and the restitution law leads the relative velocity between the two solids to be multiplied by a factor !e at the impact time:

xR (t>)!xR (t>)"!e (xR (t\)!xR (t\)).
Moreover, the conservation of the momentum provides the second equation to be able to determine post-impact velocities:

m xR (t>)#m xR (t>)"m xR (t\)#m xR (t\) .
A simple change of co-ordinates can be applied in order to express the system (14) in the form previously studied; let For this new system of variables, the equations become

x"x !x , y" m m x #x . (38)
xK #ax # x#k y"f cos ( t)#&&impact'', yK #ayR # x#k x"f cos ( t), ( 39
) x !x *0, where "(m #m )/(m #m ), "(m #m )/(m #m ), k "(m /(m # m )) ( ! ), k "(m / (m #m )) ( ! ), f "f !f and f "(m /m
) f #f . Thus the system is of the type (14); the results of the preceding study are then applicable. First of all note that k and k are proportional, so that there is always the case of the strong coupling studied in section 3. As regards the solutions of the system between two impacts, there is no work to do in order to decouple the system, since the initial system is already written in decoupled form. Thus the eigenvalues of the system are always real, and are exactly and . In this case v "1 and v "m /m . According to equations ( 34) and ( 35), the modes of the system are given by

¹ " 1, m m , ¹ " (1, 1), ¹ " (!1, 1), ¹ " ! m m , 1 ,
where "1/(1#m /m . Using these modes, the modal superposition formula is as in equation ( 36), the expression of the modal masses being given by equation [START_REF] Arnold | Me & thodes mathe & matiques de la me & canique classique[END_REF]. It is interesting to note that in general there is no reciprocity, except if m "m . Finally, the periodic responses of the system can be dealt with. First of all, according to section 3.2.2, there are (n, 0)-periodic solutions because it was assumed that x K?V '0. In the same way, calculations of section 3.2.2 can search for (n, 1)-periodic solutions. Diagrams of existence of periodic solutions versus the frequency of the external excitation are obtained. From these periodic solutions, the modal superposition formula can then be tested, plotting the di!erence between the spectral amplitude of the system's response and the nth harmonic amplitude given by equation [START_REF] Hsu | [END_REF]. For the "rst example presented, let m "1, m "0)7, "5, "13, a"0)05, e"0)9, f "20, f "18, and x K?V "14. As it was the case for the single-d.o.f. system, note in Figure 25 that the usual resonance of linear systems no longer occurs. Indeed, when the frequency of the external excitation is equal to the natural frequency of the system, the modal mass corresponding to the excited mode goes through a local maximum, so that the associated spectral amplitude does not present a particular peak. Nevertheless, spectral amplitude peaks appear for values of far away from the natural frequencies. The main peak is located in the neighbourhood of (which is one of the natural frequencies of the system after the change of variables), but it is di$cult to show a resonance in , for the peak is appreciably shifted from this frequency. From the modal point of view, the occurrence of such peaks does not correspond to a classical resonance (where the generalized natural frequency of the system is close to the frequency of external excitation); resonance can be interpreted as the locus of frequency where the generalized modal mass is minimum with respect to .

As regards the closeness of the approximation to the spectral amplitude by the nth harmonic amplitude, note, as in the single-d.o.f. case, that the "rst harmonic is not always the leading term in the amplitude of a (1, 1)-periodic response (see Figure 26). Thus, the constant coe$cient c overrides all the others for some values of . Yet if is considered to be close to the peaks in the spectral response, the "rst harmonic gives a close approximation to the total amplitude (see Table 2).

For the last example, a set of parameters close to the second example in the single-d.o.f. system is chosen: m "1, m "0)7, "1, "13, a"0)02, e"0)9, f "20, f "18, and x K?V "1. Looking at Figures 27 and 28 "rst note that amplitude peaks appear at the same frequencies for x and x , which is due to energy transmission between the two bodies through impacts. Moreover, the "rst harmonic amplitude can once again be far lower than the spectral amplitude; most of amplitude peaks in x come from A , and for x the second harmonic is often overriding.

Only the (1, 1)-periodic responses of the system have been studied. Nevertheless, it must be kept in mind that many other types of periodic solutions are possible, as was seen for the single-d.o.f. system, depending on the time period and the number of impacts per cycle. Theoretically, nothing can prevent a modal superposition formula for any (n, k)-periodic response from being written, but in practice such a response is hard to "nd by analytical means when k'1.

CONCLUSION

The feasibility of building a modal superposition formula for systems with irregular non-linearities of impact type has been investigated, imitating the procedure used in the smooth non-linear case [49,[START_REF] Lamarque | [END_REF]. The formula has been built for simple single-and two-d.o.f. systems with unilateral constraint and restitution law. The generalized modes and frequencies obtained turn out to be identical to the linear case, the non-linearity of the system being concentrated into the modal masses.

The examples considered show that the formula is valid in the case of a primary resonance for which the spectral amplitude is given by the Fourier coe$cient corresponding to the periodicity of the forced solution obtained. Nevertheless, these examples have above all illustrated the limitations to such a formulation. Firstly, the multiplicity of periodic solutions, with di!erent periodicity or number of impacts per cycle, compels several potential formulas to be built, and it is not possible to know a prori which one has to be used. Furthermore, main amplitude peaks may appear away from any a priori clearly identi"able resonance, which may be overidden by some unusual harmonics and consequently cause the modal superposition formula to fail. Therefore, it is not possible, in a general case, to build a modal superposition formula using only the usual sequence de"nition of generalized frequencies, de"nition of generalized modes and then de"nition of generalized modal masses; the non-linearities of impact type produce a limitation on the formulation of a general formula following the usual procedure.

Figure 1 .

 1 Figure 1. Single-degree-of-freedom system with k" m, c"am, g"fm.

Figure 2 .

 2 Figure 2. (1, 1)-periodic solution to the system (2) for "2)6, x "13)32968 and xR "19)36619.

Figure 3 .

 3 Figure 3. Existence of (1, 1) periodic solutions. Stable (square: x , diamond: x ) and unstable (dotted curve) solutions.

Figure 4 .

 4 Figure 4. (1, 2)-periodic solution for "1)45, x "!9)89493 and x "!20)71102.

Figure 5 .

 5 Figure 5. (3, 2)-periodic solution for "2)6, x "9)77211 and xR "35)46891.

Figure 6 .

 6 Figure 6. (4, 2)-periodic solution for "2)6, x "4)76445 and xR "47)91445.

Figure 7 .

 7 Figure 7. Existence of (1, 2) periodic solutions. Stable (square: x , diamond: x ) and unstable (dotted curve) solutions.

Figure 8 .

 8 Figure 8. Existence of (3, 2)-periodic solutions: stable (solid curve) and unstable solutions (dotted curve): (a) initial displacements; (b) initial velocities.

Figure 9 .

 9 Figure 9. Existence of (4, 2)-periodic solutions: stable (solid curve) and unstable solutions (dotted curve): (a) initial displacements; (b) initial velocities.

Figure 10 .

 10 Figure 10. First-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (1, 1)-periodic solutions.

Figure 11 .

 11 Figure 11. Fourier coe$cients of the (1, 1)-periodic responses.

Figure 12 .

 12 Figure 12. First-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (1, 2)-periodic solutions.

Figure 13 .

 13 Figure 13. Fourier coe$cients of the (1, 2)-periodic responses.

Figure 14 .

 14 Figure 14. Third-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (3, 2)-periodic solutions.

Figure 15 .

 15 Figure 15. Fourier coe$cients of the (3, 2)-periodic responses.

Figure 16 .

 16 Figure 16. Fourth-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (4, 2)-periodic solutions.

Figure 17 .

 17 Figure 17. Fourier coe$cients of the (4, 2)-periodic responses.

Figure 18 .

 18 Figure 18. Fourier coe$cients of the (1, 1)-periodic responses.

3. 1 . 1 .

 11 Analytical solution of the system 3.1.1.1. Decoupling equations. The system (14) can be decoupled in order to obtain explicit solutions for x and y. If v "

  four unknowns are obtained b

Figure 19 .

 19 Figure 19. First-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (1, 1)-periodic responses: (a) First degree of freedom x; (b) second degree of freedom y.

Figure 20 .

 20 Figure 20. First-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (1, 1)-periodic responses: (a) First degree of freedom x; (b) second degree of freedom y.

Figure 21 .

 21 Figure 21. Mechanical model of two rigid oscillating bodies colliding.

Figure 22 .

 22 Figure 22. (1, 1)-periodic solution for "4, x "!7)90482, xR "!2)07935, x "11)20441, and xR "15)14127: absolute displacement.

Figure 23 .

 23 Figure 23. Existence of (1, 0)-periodic solutions: (a) initial displacements for x (circle) and x (x-mark); (b) initial velocities for x (circle) and x (x-mark).

Figure 24 .

 24 Figure 24. Existence of (1, 1)-periodic solutions: (a) initial displacements for x (circle) and x (x-mark); (b) initial velocities for x (circle) and x (x-mark).

Figure 25 .

 25 Figure 25. First-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (1, 1)-periodic solutions: (a) "rst degree of freedom x ; (b) second degree of freedom x .

Figure 26 .

 26 Figure 26. Fourier coe$cients of the (1, 1)-periodic responses: (a) "rst degree of freedom x ; (b) second degree of freedom x .

Figure 27 .

 27 Figure 27. First-harmonic amplitude (dotted curve) and modal mass (solid curve) of the (1, 1)-periodic solutions: (a) "rst degree of freedom x ; (b) second degree of freedom x .

Figure 28 .

 28 Figure 28. Fourier coe$cients of the (1, 1)-periodic responses: (a) "rst degree of freedom x ; (b) second degree of freedom x .

TABLE 1

 1 Di+erence between Fourier amplitude and nth harmonic amplitude

	Periodic	Di!erence (%)	Di!erence (%)
	solution	stable or unstable solution	stable solution
	(1, 1)	36)65	36)65
	(1, 2)	67)89	67)89
	(3, 2)	15)56	13)96
	(4, 2)	40)89	28)96

  3.2.3.2. Generalized masses and modal superposition. It is now possible to consider a (n, k)-periodic solution and to calculate the Fourier coe$cients of x

	and x	. The
	calculation of c H only the recursive relation (29) changes. In this case ( ) and c ( ) is similar to the one carried out in the weak coupling case: H

TABLE 2

 2 Di+erence between the whole spectral amplitude and the amplitude of nth harmonic

	Degree of	Di!erence (in %)	Di!erence (in %)
	freedom	whole range	neighbourhood of the peak
	x	77)31	12)82
	x	82)75	4)89