Jacques M Bahi

Christophe Guyeux

Antoine Perasso
email: antoine.perasso@univ-fcomte.fr

Relaxing the Hypotheses of Symmetry and Time-Reversibility in Genome Evolutionary Models

Keywords: Genome evolutionary models, stochastic processes, discrete dynamical system, matrix calculus, nucleotides mutations. 2012 Mathematics Subject Classification: 15A51, 15A16, 60G99, 92D10, 92D15, 92D20

Various genome evolutionary models have been proposed these last decades to predict the evolution of a DNA sequence over time, essentially described using a mutation matrix. By essence, all of these models relate the evolution of DNA sequences to the computation of the successive powers of the mutation matrix. To make this computation possible, hypotheses are assumed for the matrix, such as symmetry and time-reversibility, which are not compatible with mutation rates that have been recently obtained experimentally on genes ura3 and can1 of the Yeast Saccharomyces cerevisiae. In this work, authors investigate systematically the possibility to relax either the symmetry or the time-reversibility hypothesis of the mutation matrix, by investigating all the possible matrices of size 2 ˆ2 and 3 ˆ3. As an application example, the experimental study on the Yeast Saccharomyces cerevisiae has been used in order to deduce a simple mutation matrix, and to compute the future evolution of the rate purine/pyrimidine for ura3 on the one hand, and of the particular behavior of cytosines and thymines compared to purines on the other hand.

Introduction

Due to mutations or recombination, some variations occur in the frequency of each codon, and these codons are thus not uniformly distributed into a given genome. Since the late '60s, various genome evolutionary models have been proposed to predict the evolution of a DNA sequence as generations pass. Mathematical models allow the prediction of such an evolution, in such a way that statistical values observed in current genomes can be at least partially recovered from hypotheses on past DNA sequences. Moreover, it can be attractive to study the genetic patterns (blocs of more than one nucleotide: dinucleotides, trinucleotides...) that appear and disappear depending on mutation parameters.

A first model for genomes evolution has been proposed in 1969 by Thomas Jukes and Charles Cantor [START_REF] Jukes | Evolution of protein molecules[END_REF]. This first model is very simple, as it supposes that each nucleotide has the probability m to mutate to any other nucleotide, as described in the following mutation matrix,

¨˚m m m m ˚m m m m ˚m m m m ˚‹ ‹ ' .
In that matrix, the nucleotides are ordered as pA, C, G, T q, so that for instance the coefficient in row 3, column 2 represents the probability that the nucleotide G mutates into a C during the next time interval, i.e., P pG Ñ Cq. As diagonal elements can be deduced by the fact that the sum of each row must be equal to 1, they are omitted here. This first attempt has been followed up by Motoo Kimura [START_REF] Kimura | A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences[END_REF], who has reasonably considered that transitions (A ÐÑ G and T ÐÑ C) should not have the same mutation rate than transversions (A ÐÑ T , A ÐÑ C, T ÐÑ G, and C ÐÑ G), this model being refined by Kimura in 1981, with three constant parameters to make a distinction between natural A ÐÑ T , C ÐÑ G and unnatural transversions, leading to:

¨˚c a b c ˚b a a b ˚c b a c ˚‹ ‹ ' .
Joseph Felsenstein [START_REF] Felsenstein | A view of population genetics[END_REF] has then supposed that the nucleotides frequency depends on the kind of nucleotide A,C,T,G. Such a supposition leads to a mutation matrix of the form:

¨˚πC πG πT πA ˚πG πT πA πC ˚πT πA πC πG ˚‹ ‹ '
with πA, πC , πG, and πT denoting the frequency of occurance of each nucleotide, respectively. Masami Hasegawa, Hirohisa Kishino, and Taka-Aki Yano [START_REF] Hasegawa | Dating of the human-ape splitting by a molecular clock of mitochondrial dna[END_REF] have generalized the models of [START_REF] Kimura | A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences[END_REF] and [START_REF] Felsenstein | A view of population genetics[END_REF], introducing in 1985 the following mutation matrix: These efforts have been continued by Tamura, who proposed in [START_REF] Tamura | Estimation of the number of nucleotide substitutions when there are strong transitiontransversion and g+c-content biases[END_REF][START_REF] Tamura | Estimation of the number of nucleotide substitutions in the control region of mitochondrial DNA in humans and chimpanzees[END_REF] a simple method to estimate the number of nucleotide substitutions per site between two DNA sequences, by extending the model of [START_REF] Kimura | A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences[END_REF]. The idea is to consider a two-parameter method, for the case where a GC bias exists. Let us denote by πGC the frequency of this dinucleotide motif. Tamura supposes that πG " πC " πGC 2 and πA " πT " 1 ´πGC 2 , which leads to the following rate matrix:

¨˚απC
Mutation ura3 can1 T Ñ C 4 4 T Ñ A 14 9 T Ñ G 5 5 C Ñ T 16 20 C Ñ A 40 21 C Ñ G 11 9 A Ñ T 8 4 A Ñ C 6 5 A Ñ G 0 1 G Ñ T 28 20 G Ñ C 9 12 G Ñ A
¨˚κp1 ´πGC q{2 p1 ´πGC q{2 p1 ´πGC q{2 κπGC {2 ˚πGC {2 πGC {2 p1 ´πGC q{2 p1 ´πGC q{2 ˚κp1 ´πGC q{2 πGC {2 πGC {2 κπGC {2 ˚‹ ‹ ' .
All these models are special cases of the GTR model [START_REF] Yang | Estimating the pattern of nucleotide substitution[END_REF], in which the mutation matrix has the form (using obvious notations): ¨˚fAC πC fAGπG fAT πT fAC πA ˚fCGπG fCT πT fAGπA fCGπC ˚πT fAT πA fCT πC πG ˚‹ ‹ ' .

Non-reversible and non-symmetric models have, for their part, been considered in practical inferences since at least a decade for phylogenetic studies, see for instance [START_REF] Klosterman | XRate: a fast prototyping, training and annotation tool for phylo-grammars[END_REF][START_REF] Boussau | Efficient likelihood computations with nonreversible models of evolution[END_REF][START_REF] Yap | Rooting a phylogenetic tree with nonreversible substitution models[END_REF]. As they are more regarded for their interest in practical inference investigations than on the theoretical side, they will not be developed in this article. Due to mathematical complexity, matrices theoretically investigated to model evolution of DNA sequences are thus limited either by the hypotheses of symmetry and time-reversibility or by the desire to reduce the number of parameters under consideration. These hypotheses allow their authors to solve theoretically the DNA evolution problem, for instance by computing directly the successive powers of their mutation matrix. However, one can wonder whether such restrictions on the mutation rates are realistic. Focusing on this question, we used in [START_REF] Bahi | Predicting the evolution of gene ura3 in the yeast saccharomyces cerevisiae[END_REF] a recent research work of Lang and Murray [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF], in which the per-base-pair mutation rates of the Yeast Saccharomyces cerevisiae have been experimentally measured (see Table 1), allowing us to calculate concrete mutation matrices for genes ura3 and can1. We deduced in [START_REF] Bahi | Predicting the evolution of gene ura3 in the yeast saccharomyces cerevisiae[END_REF] that none of the existing genomes evolution models can fit such mutation matrices, implying the fact that some hypotheses must be relaxed, even if this relaxation implies less ambitious models: current models do not match with what really occurs in concrete genomes, at least in the case of this yeast. Having these considerations in mind, the data obtained by Lang and Murray have been used in [START_REF] Bahi | Predicting the evolution of gene ura3 in the yeast saccharomyces cerevisiae[END_REF] in order to predict the evolution of the rates or purines and pyrimidines in the particular case of ura3. Mathematical investigations and numerical simulations have been proposed, focusing on this particular gene and its associated matrix of size 2ˆ2 (purines vs. pyrimidines), and of size 3ˆ3 (cytosines and thymines compared to purines). Instead of focusing on two particular matrices, this extension of [START_REF] Bahi | Predicting the evolution of gene ura3 in the yeast saccharomyces cerevisiae[END_REF] investigates systematically all the possible mutation matrices of sizes 2 ˆ2 and 3 ˆ3. Thus, the study is finalized in this article, by investigating all the possible cases, and discussing about their mathematical and biological relevance.

The remainder of this research work is organized as follows. First of all the case of mutation matrices of size 2 ˆ2 is recalled in Section 2 and applied to the ura3 gene taking into account purines and pyrimidines mutations. A simulation is then performed to compare this non reversible model to the classical symmetric Cantor model. The next sections deal with all the possible 6-parameters models of size 3 ˆ3. In Section 3, a complete theoretical study is led encompassing all the particular situations, whereas in Section 4 an illustrative example focusing on the evolution of the purines, cytosines, and thymines triplet is given for ura3. We finally conclude this work in Section 5.

General Model of Size ˆ2

In this section, a first general genome evolution model focusing on purines versus pyrimidines is proposed, to illustrate the method and as a pattern for further investigations. This model is applied to the case of the yeast Saccharomyces cerevisiae.

A convergence result

Let R and Y denote respectively the occurrence frequency of purines and pyrimidines in a sequence of nucleotides, and M " ˆa b c d ˙the associated mutation matrix, with a " P pR Ñ Rq, b " P pR Ñ Y q, c " P pY Ñ Rq, and

d " P pY Ñ Y q satisfying # a `b " 1, c `d " 1, (2.1)
and thus M " ˆa 1 ´a c 1 ´c ˙. The initial probability is denoted by P0 " pR0 Y0q, where R0 and Y0 denote respectively the initial frequency of purines and pyrimidines. So the occurrence probability at generation n is Pn " P0M n , where Pn " pRpnq Y pnqq is a probability vector such that Rpnq (resp. Y pnq) is the rate of purines (resp. pyrimidines) after n generations. The following theorem states the time asymptotic behavior of the probabilit Pn.

We recall the following result was proved in [START_REF] Bahi | Predicting the evolution of gene ura3 in the yeast saccharomyces cerevisiae[END_REF]: thm 2.1. Consider a DNA sequence under evolution, whose mutation matrix is M " ˆa 1 ´a c 1 ´c ẇith a " P pR Ñ Rq and c " P pY Ñ Rq.

• If a " 1, c " 0, then the frequencies of purines and pyrimidines do not change as the generation pass.

• If a " 0, c " 1, then these frequencies oscillate at each generation between pR0 Y0q (even generations) and pY0 R0q (odd generations).

• Else the value Pn " pRpnq Y pnqq of purines and pyrimidines frequencies at generation n is convergent to the following limit:

lim nÑ8 Pn " 1 c `1 ´a `c 1 ´a ˘.
rem 2.1. Note that the case a ‰ 1 ´c, resp. a ‰ c, translates the non symmetry property, resp. the time reversibility property.

Numerical Application

For numerical application, we will consider mutations rates in the ura3 gene of the Yeast Saccharomyces cerevisiae, as obtained by Gregory I. Lang and Andrew W.Murray [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF] and summed up in Table 1.

2 ˆ2. So c " P pY Ñ Rq " m ˆ1 ´743 14362 ˙« 2.897 ˆ10 ´7.
As a consequence the purine/pyrimidine mutation matrix that corresponds to the values of where m " 3.0552 ˆ10 ´7 as mentionned previously.

Using the value of m for the ura3 gene leads to 1 ´a " 2.83391 ˆ10 ´7 and c " 2.89714 ˆ10 ´7, which can be used in Theorem 2.1 to conclude that the rate of pyrimidines is convergent to 49.45% whereas the rate of purines converge to 50.55%. Numerical simulations using data published in [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF] are given in Figure 2, leading to a similar conclusion.

A First Genomes Evolution Model of size ˆhaving 6 Parameters without Time-reversibility hypothesis

In order to investigate the evolution of the frequencies of cytosines and thymines in the gene ura3, a model of size 3 ˆ3 compatible with real mutation rates of the yeast Saccharomyces cerevisiae is now presented.

Formalization

Let us consider a line of yeasts where a given gene is sequenced at each generation, in order to clarify explanations. The n´th generation is obtained at time n, and the frequences of purines, cytosines, and thymines at time n are respectively denoted by PRpnq, PC pnq, and PT pnq.

Let a be the probability that a purine is changed into a cytosine between two generations, that is: a " P pR Ñ Cq. Similarly, denote by b, c, d, e, f the respective probabilities: P pR Ñ T q, P pC Ñ Rq, P pC Ñ T q, P pT Ñ Rq, and P pT Ñ Cq. Contrary to existing approaches, P pR Ñ Cq is not supposed to be equal to P pC Ñ Rq, and the same statement holds for the other probabilities. For the sake of simplicity, we will suppose in all that follows that a, b, c, d, e, f are not time dependent.

Let

M " ¨1 ´a ´b a b c 1 ´c ´d d e f 1 ´e ´f '
be the mutation matrix associated to the probabilities mentioned above, and Pn the vector of occurrence, at time n, of each of the three kind of nucleotides. In other words, Pn " pPRpnq PC pnq PT pnqq. Under that hypothesis, Pn is a probability vector: @n P N,

• PRpnq, PC pnq, PT pnq P r0, 1s,

• PRpnq `PC pnq `PT pnq " 1,

Let P0 " pPRp0q PC p0q PT p0qq P r0, 1s 3 be the initial probability vector. We have obviously:

PRpn `1q " PRpnqP pR Ñ Rq `PC pnqP pC Ñ Rq `PT pnqP pT Ñ Rq, with similar equalities for PC pn `1q and PT pn `1q so that

Pn " Pn´1M " P0M n . ( 3.1)
In all that follows we wonder if, given the parameters a, b, c, d, e, f as in [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF], one can determine the frequency of occurrence of any of the three kind of nucleotides when n is sufficiently large, in other words if the limit of Pn is accessible by computations.

Resolution

This section, that is a preliminary of the convergence study, is devoted to the determination of the powers of matrix M in the general case and some particular situations

Determination of M n in the general case

The characteristic polynomial of M is equal to χM pxq " x 3 `ps ´3qx 2 `pp ´2s `3qx ´1 `s ´p " px ´1q `x2 `ps ´2qx `p1 ´s `pq ˘,

where

s " a `b `c `d `e `f,
p " ad `ae `af `bc `bd `bf `ce `cf `de, detpM q " 1 ´s `p.

The discriminant of the polynomial of degree 2 in the factorization of χM is equal to ∆ " ps 2q 2 ´4p1 ´s ´pq " s 2 ´4p. Let x1 and x2 the two roots (potentially complex or equal) of χM , given by x1 " ´s `2 ´as 2 ´4p 2 and x2 " ´s `2 `as 2 ´4p 2 .

(3.2)

Let n P N, n ě 2. As χM is a polynomial of degree 3, a division algorithm of X n by χM pXq leads to the existence and uniqueness of two polynomials Qn and Rn, such that

X n " QnpXqχ2pXq `RnpXq, (3.3)
where the degree of Rn is lower than or equal to the degree of χM , i.e., RnpXq " anX 2 `bnX `cn with an, bn, cn P R for every n P N. By evaluating (3.3) in the three roots of χM , we find the system

$ & % 1 " an `bn `cn x n 1 " anx 2 1 `bnx1 `cn x n 2 " anx 2 2 `bnx2 `cn
This system is equivalent to

$ & % cn `bn `an " 1 bnpx1 ´1q `anpx 2 1 ´1q " x n 1 ´1 bnpx2 ´1q `anpx 2 2 ´1q " x n 2 ´1
If we suppose that x1 ‰ 1, x2 ‰ 1, and x1 ‰ x2, then standard algebraic computations give

$ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % an " 1 x2 ´x1 " x n 2 ´1 x2 ´1 ´xn 1 ´1 x1 ´1 ȷ , bn " x1 `1 x1 ´x2 x n 2 ´1 x2 ´1 `x2 `1 x2 ´x1 x n 1 ´1 x1 ´1 ,
cn " 1 ´an ´bn.

Using for i " 1, 2 and n P N the following notation,

Xipnq " x n i ´1 xi ´1 , ( 3.4)
and since x2 ´x1 " ? ∆, the system above can be rewritten as (3.5)

By evaluating (3.3) in M and due to the theorem of Cayley-Hamilton, we finally have for every integer n ě 1,

M n " anM 2 `bnM `cnI3, (3.6)
where I3 is the identity matrix of size 3, an, bn, and cn are given by (3.5), and M 2 is given by

M 2 "
¨a2 `2ab `ac ´2a ´a2 ´ab ´ac ´ab `ad ´b2 `b2 `be ´2b `1 ´ad `2a `bf ´be ´bf `2b ´ac ´bc ´c2 ac `c2 `2cd ´2c bc ´cd ´d2 ´cd `2c `de `d2 `df ´2d `1 ´de ´df `2d ´ae ´be `cf

ae ´cf ´df be `df `e2 `2ef ´e2 ´ef `2e ´ef ´f 2 `2f ´2e `f 2 ´2f `1 ‹ ‹ ‹ ‹ ‹ ‹ ' .

Determination of M n in particular situations

Formulations of (3.5) only hold for x1 ‰ x2, x1 ‰ 1, and x2 ‰ 1. We now investigate these latter cases.

Preliminaries Let us firstly remark that, as the mutation matrix M is stochastic, we have necessarily 0 ď a `b ď 1, 0 ď c `d ď 1, and 0 ď e `f ď 1. These inequalities imply that s P r0, 3s. Consequently from the definition of p one can check that p " ad `ape `f q `bpc `dq `bf `cpe `f q `de ď ad `a `b `bf `c `de ď s, as each parameter is in r0, 1s. To sum up, Note that, as we deal with a stochastic process, the module of the eigenvalues of M are smaller than 1, so |x1| ď 1 and |x2| ď 1.

0 ď p ď s ď 3. ( 3
Suppose that x 1 " 1 Then ´s " a s 2 ´4p ðñ s " p " 0. So a " b " c " d " e " f " 0, and the mutation matrix is equal to the identity of size 3. Conversely, if a " b " c " d " e " f " 0, then x1 " 1.

In that situation, the system does not evolve.

Suppose that x 2 " 1 (and x 1 ‰ 1) Then s " a s 2 ´4p ðñ p " 0. In that situation, x1 " 1 ´s and 1 is root of multiplicity 2 of χ2, whereas x1 " 1 ´s is its third root. As the case x1 " 1 has already been regarded, we can consider that s ‰ 0. Using (3.3), These facts lead to the following system:

$ & % 1 " an `bn `cn, n " 2an `bn, p1 ´sq n " p1 ´sq 2 an `p1 ´sqbn `cn.

Standard computations then give the following formula:

$ ' ' ' ' & ' ' ' ' % an " ´1 `sn `p1 ´sq n s 2 ,
bn " p3 ´sq `ps 2 ´2sqn `ps ´3qp1 ´sq n s , cn " ps ´1qp2s ´1q ´sps ´1q 2 n ´ps 2 ´3s `1qp1 ´sq n s 2 .

(3.9)

Case x 1 " x 2 ‰ 1 (∆ " 0) Then (3.8) implies that x1 " 1´s{2 P " ´1 2 , 1 ˘.
From a differentiation of (3.3) one deduces that x1 satisfies the following system for every n P N

˚, $ & % 1 " an `bn `cn x n 1 " anx 2 1 `bnx1 `cn nx n´1 1 " 2anx1 `bn Standard algebraic computations give, since x1 ‰ 1, $ ' ' ' ' ' ' & ' ' ' ' ' ' % an " n x n´1 1 x1 ´1 ´X1pnq x1 ´1
bn " X1pnq ´anpx1 `1q

cn " 1 ´an ´bn

(3. 10
)
where X1pnq is defined in (3.4).

Convergence study

Convergence study in the general case

We suppose in this section that x1 ‰ x2, x1 ‰ 1, and x2 ‰ 1. So formulations of (3.5) hold for an, bn, and cn. We split the study convergence in several sub-cases, that are the objects of Theorems

Convergence study in particular situations

The case where x1 " 1 has already been discussed, it implies that a " b " c " d " e " f " 0, and so the system does not evolve. The other particular situations are invastigated in the two following theorems.

thm 3.4. Suppose that x2 " 1 and x1 ‰ 1 (or equivalently p " 0). Then the system is well formulated if and only if M 2 `sps ´2qM ´ps ´1q 2 I3 ‰ 0. In that situation, we have:

• either s Ps0, 2r, and so pPRpnq PC pnq PT pnqq ÝÑ pPRp0q PC p0q PT p0qq ˆ1 s 2 r´M 2 `sp3 śqM `ps ´1qp2s ´1qI3s.

• or s " 2, and so pPRp2nq PC p2nq PT p2nqq ÝÑ pPRp0q PC p0q PT p0qq whereas pPRp2n `1q PC p2n `1q PT p2n `1qq pPRp0q PC p0q PT p0qq ˆp´2M 2 `4M `2I3q.

Proof. Using (3.9), we can deduce that M n is equal to: anM 2 `bnM `cnI3 " n s rM 2 `sps ´2qM ´ps ´1q 2 I3s `1 s 2 p1 ´sq n rM 2 `sps ´3qM ´ps 2 ´3s `1qI3s `1 s 2 r´M 2 `sp3 ´sqM `ps ´1qp2s ´1qI3s.

Several cases can be deduced from this equality.

Application in Concrete Genomes Prediction

We consider another time the numerical values for mutations published in [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF]. Gene ura3 of the Yeast Saccharomyces cerevisiae has a mutation rate of 3.80 ˆ10 ´10 /bp/generation [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF]. As this gene is constituted by 804 nucleotides, we can deduce that its global mutation rate per generation is equal to m " 3.80 ˆ10 ´10 ˆ804 " 3.0552 ˆ10 ´7. Let us compute the values of a, b, c, d, e, and f . The first line of the mutation matrix is constituted by 1 ´a ´b " P pR Ñ Rq, a " P pR Ñ T q, and b " P pR Ñ Cq. P pR Ñ Rq takes into account the fact that a purine can either be preserved (no mutation, probability 1 ´m), or mutate into another purine (A Ñ G, G Ñ A). As the generations pass, authors of [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF]

Conclusion

In this document, a formulation of the non symmetric discrete model of size 2 ˆ2 has been proposed, which studies a DNA evolution taking into account purines and pyrimidines mutation rates. A simulation has been performed, to compare the proposal to the well known Jukes and Cantor model. Then all non-symmetrical models of size 3x3 that have 6 parameters have been studied theoretically. They have been tested with numerical simulations, to make a distinction between cytosines and thymines in the former proposal. These two models still remain generic, and can be adapted to a large panel of applications, replacing either the couple (purines, pyrimidines) or the tuple (purines, cytosines, thymines) by any categories of interest.

Remark that the ura3 gene is not the unique example of a DNA sequence of interest such that none of the existing nucleotides evolution models cannot be applied due to a complex mutation matrix. For instance, a second gene called can1 has been studied too by the authors of [START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF]. Similarly to gene ura3, usual models cannot be used to predict the evolution of can1, whereas a study following a same canvas than what has been proposed in this research work can be realized.

In future work, biological consequences of the results produces by these models will be systematically investigated. Then, the most general non symmetric model of size 4 will be regarded in some particular cases taken from biological case studies, and the possibility of mutations non uniformly distributed will then be regarded. Finally, this 4 ˆ4 general case will be investigated using Perron-Frobenius based approaches instead of using methods directly inspired by linear algebra, in order to obtain the most global results on mutation matrices.

Figure 2 :

 2 Figure 2: Prediction of purine/pyrimidine evolution of ura3 gene in general model of size

Table 1 :

 1 Summary of sequenced ura3 and can1 mutations[START_REF] Lang | Estimating the per-base-pair mutation rate in the yeast saccharomyces cerevisiae[END_REF]

		26	40
	Transitions	46	65
	Transversions	121	85

  They have measured phenotypic mutation rates, indicating that the per-base pair mutation rate at ura3 is m " 3.0552 ˆ10 ´7/generation for the whole gene.For the majority of Yeasts they studied, ura3 is constituted by 804 bp: 133 cytosines, 211 Using these values in the historical model of Jukes and Cantor[START_REF] Jukes | Evolution of protein molecules[END_REF], we obtain the evolution depicted in Figure1.2.1 allows us to compute the limit of the rates of purines and pyrimidines:

			0.58				Nucleotides Evolution Rates
								Purine
			0.56				Pyrimidine
			0.54
			0.52
		Rate	0.50
	thymines, 246 adenines, and 214 guanines. So R0 " 1 2 3 4 5 Time (1 unit = 400000000 generations) 246 `214 804 6 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 Rate Nucleotides Evolution Rates 0 1 2 3 4 5 6 Time (1 unit = 1 mutation) 0.42 0.44 0.46 0.428. 0 0.48	« 0.572, and Y0 " 7 8 9 Purine Pyrimidine 7 8 9	133 `211 804	«
	Figure 1: Prediction of purine/pyrimidine evolution of ura3 gene in symmetric Cantor
	model.
	Computation of probability a. P pR	Ñ	Rq	"	p1	´mq
	`P pA Ñ Gq	PApnq PApnq `PGpnq	`P pG Ñ Aq	PGpnq PApnq `PGpnq	. The use of Table 1 and the
	hypothesis that the base frequencies have already reached their steady states implies that
	a " p1 ´mq `ˆm	0 46 `121	˙ˆ246 804 246 804 `214 804	`ˆm	26 46 `121	˙ˆ214 804 246 804 `214 804	. We thus obtain that
	a " 1	´17814m 19205	« 0.999999716.
	Computation of probability c. Similarly, P pY Ñ Y q " p1 ´mq `P pC Ñ T q Cq PT PC `PT " p1´mq`m 16 46 `121 ˆ133 133 `211 `m 4 46 `121 ˆ211 133 `211 PC `PT PC " 1´m`m `P pT Ñ 743 14362 .

Theorem

Table 1

 1

	is:	M " m	¨1391 19205 13619	17814 19205 743	‹ ‹ ' .	(2.2)
			14362	14362

  The case |x1| ‰ 1 and |x2| " 1 necessarilly implies that x2 " 1, which is in contradiction with the assumptions made in preamble of Section 3.3.1. Proof. Suppose that |x1| " |x2|, but x1, x2 P CzR. Then x1 and x2 are complex and conjugate, of the form x1 " e ´iθ , x2 " e iθ , with θ ı 0rπs. So x1 ´x2 " ? ∆ " e ´iθ ´eiθ " ´2i sinpθq, and

	and finally, rem 3.1. thm 3.3. If |x1| " |x2|, but x1, x2 P CzR, then pPRpnq PC pnq PT pnqq " pPRp0q PC p0q PT p0qq 2 sin `nθ 2 ˘sin ´pn´2qθ 2 sinpθq ¯cos `θ 2 bn " sin `θ 2 ˘.	panM
	2 `bnM `cnI3q, where As cn " 1 ´an ´bn, we have:
	• an "	´sin `nθ 2 ˘sin ´pn´1qθ 2 `θ 2 ˘sinpθq cn " 1 sin , ´sin `nθ 2 ˘sin ´pn´3qθ 2 sin `θ 2 ˘. sinpθq
	• bn "	2 sin `nθ 2 ˘sin ´pn´2qθ 2 sin `θ 2	sinpθq ˘, ¯cos `θ 2
	• cn " 1	2 ˘sin ´pn´3qθ 2 ´sin `nθ sin `θ 2 ˘. sinpθq
	with e ´iθ " x1.
	• PRpnq ÝÑ • PC pnq ÝÑ	ce `cf `de an p ´bf `df ae `af `bf p ´bf `df 2i sinpθq an " "	X2pnq ´X1pnq ? ∆ e ´inθ ´1 e ´iθ ´1 ´einθ ´1 X1pnq ´X2pnq " 2i sinpθq e iθ ´1
	• PT pnq ÝÑ	ad `bc `bd p ´bf `df	"	e ´in θ 2 e ´i θ 2	e ´in θ 2 ´ein θ 2 e ´i θ 2 ´ei θ 2	2 ´ein θ e i θ 2	e in θ 2 ´e´in θ 2 e i θ 2 ´e´i θ 2
	Proof. If |x1| ă 1 and |x2| ă 1 then Xipnq ÝÑ an ÝÑ 1 ? " e ´i pn´1qθ 2 ´2i sin `nθ 1 1 ´xi for i " 1, 2 and so ˙. 2 2i sin `θ 2 ˘´e i pn´1qθ 2i sin `nθ 2 2 " 2i sin `θ 2 ∆ ˆ1 1 ´x2 ´1 1 ´x1 Denote by a8 this limit. We have sin `nθ 2 sin `θ 2 ˘´e ´i pn´1qθ 2 2 ´ei pn´1qθ ¯.
	Finally, and finally		a8 "	?	x2 ´x1 ∆p1 ´x2qp1 ´x1q an " ´sin `nθ " p1 ´x2qp1 ´x1q 1 2 ˘sin ´pn´1qθ 2 `θ 2 ˘sinpθq	" sin 1 s `?∆ 2 s ´?∆ 2 .	,
	a8 " Similarly, bn " X1pnq ´anpx1 `1q satisfies Similarly, ? ∆bn " px2 `1qX1pnq ´px1 `1qX2pnq 4 s 2 ´∆ " 1 . p
	bn ÝÑ ´2i sinpθqbn " `eiθ `1˘e ´i pn´1qθ 1 1 ´x1 The following computations 2 sin `nθ ´x1 2 sin `1 p `θ 2 ˘´`e ´iθ `1˘e i pn´1qθ . 2	sin " sin `nθ 2 `θ 2
						1 1 ´x1 sin `nθ " 2 `θ 2 ˘"e 2 s `?∆ sin ´i pn´3qθ " 2	2ps ´?∆q s 2 ´∆ " `e´i pn´1qθ 2 ´ei pn´3qθ s ´?∆ 2p , 2	2 ´ei pn´1qθ	ı
			bn		"	x1 sin `nθ 2 sinpθqsin p `1 `θ 2 ˘´sin ´pn´3qθ " ´s `4 ´?∆ 2p 2 ¯`sin ´pn´1qθ , 2	¯¯.

3.1-3.5. thm 3.1. Suppose that |x1| ă 1 and |x2| ă 1. Then the frequencies PRpnq, PC pnq, and PT pnq of occurrence at time n of purines, cytosines, and thymines in the considered gene, converge to the following values:

•

  If s Ps0, 2r, then M n is bounded if and only if M 2 `sps´2qM ´ps´1q 2 I3 " 0. In that condition, If s " 2, then another time M n is bounded if and only if M 2 `sps ´2qM ´ps ´1q 2 I3 " 0. In that condition,M 2n ÝÑ I3, whereas M 2n`1 ÝÑ ´2M 2 `4M `2I3. • Finally, if s ą 2, then as s " a `b `c `d `e `fand a, b, c, d, e, f P r0, 1s, we have necessarily at least three coefficients in a, b, c, d, e, f that are non zero. So at least one product in abc, abd, abe, abf, acd, ace, acf, ade, adf, aef, bcd, bce, bcf, bde, bdf, bef, cde, cdf, cef, def is strictly positive. This is impossible, as p " ad `ae `af `bc `bd `bf `ce `cf `de is equal to 0.Using these values in (3.6), we can determine the limit of M n , which is a8M 2 `b8M `c8I3, where I3 is the identity matrix of size 3. All computations done, we find

					s 2
				b8 "	´2x1 p1 ´x1q 2 " 4	s	´2 s 2
				c8 "	2x1 p1 ´x1q 2 `1 " ˆ1 ´1	´2 s	˙2
				M n ÝÑ	s 2 4	M31 M32 M33 M21 M22 M23 ¨M11 M12 M13	'
	with M11 "	s 2 4	´p `ce
	M n ÝÑ	1 s 2 r´M 2 `sp3 ´sqM `ps ´1qp2s ´1qI3s.
	• thm 3.5. Suppose that x1 " x2 ‰ 1 (or equivalently s 2 " 4p). Then the probabilities PRpnq, PC pnq,
	and PT pnq of occurrence at time n of a purine, cytosine, and thymine on the considered nucleotide,
	converge to the following values:
	• PRpnq ÝÑ	4 s 2 pce `cf `deq,

`cf `de, M12 " ae `af `bf , M13 " ad `bc `bd, M21 " ce `cf `de,

M22 "

s 2 4 ´p `ae `af `bf , M23 " ad `bd `bc, M31 " ce `de `cf , M32 " ae `af `bf , and

M33 " s 2

4

´p `ad `bc `bd. However, since x1 " x2, we have ∆ " s 2 ´4p " 0 and so

M n ÝÑ 4 s 2

¨ce `cf `de ae `af `bf ad `bc `bd ce `cf `de ae `af `bf ad `bc `bd ce `cf `de ae `af `bf ad `bc `bd ',

  have counted 0 mutations of kind A Ñ G, and 26 mutations of kind G Ñ A. Similarly, there were 28 mutations G Ñ T and 8: A Ñ T , so 36: R Ñ T . Finally, 6: A Ñ C and 9: G Ñ C lead to 15: R Ñ C mutations. The total of mutations to consider when evaluating the first line As x1 « 0.9999685 P r0, 1s and x2 « 0.9999686 P r0, 1s, we have, due to Theorem 3.1: So PRpnq ÝÑ 0.549, PC pnq ÝÑ 0.292, and PT pnq ÝÑ 0.159. Simulations corresponding to this example are given in Fig. 3.

						et al.; BJMCS, 5(4), 439-455, 2015; Article no.BJMCS.2015.031
				0.6		Nucleotides Evolution Rates	Purine
						Cytosine
				0.5		Thymine
			Rate	0.4
				0.3
				0.2
				0.0	0.5	1.0 Time (1 unit = 1 mutation) 1.5	2.0	2.5
	p1 ´mq , and f " « 8.134 ˆ10 ´7, and p " `m 26 77 4m 23 , e " 51m 67 16m . 67 205m , d " 77 207488m 2 , a " 36m 77 « , 118657 ą 0, x1 " 1 19m 23 In that situation, s " a `b `c `d `e `f " and b " 15m 77 . A similar reasoning leads to c " 1.632 ˆ10 ´13 . So ∆ " s 2 ´4p " 854221m 2 9136589 ´m 2 ˆ205 77 `c 854221 9136589 ˙, and x2 "
	1	´m 2 ˙. • PRpnq ÝÑ ˆ205 77 ´c 854221 9136589 ce `cf `de p ´bf `df
		• PC pnq ÝÑ	ae `af `bf p ´bf `df
		• PT pnq ÝÑ	ad `bc `bd p ´bf `df
	Using the data of [12], we find that PRp0q " PT p0q " 211 804 « 0.263.	460 804	« 0.572, PC p0q "	133 804	« 0.165, and

Figure 3: Prediction of evolution concerning the purine, thymine, and cytosine rates in ura3. Non-symmetric Model of size 3 ˆ3.

is so equal to 77. All these considerations lead to the fact that 1 ´a ´b "

Using the latter values in (3.6), we can determine the limit of M n , which is a8M 2 `b8M `c8I3. All computations done, we find the following limit for M n , 1 p ´bf `df ¨ce `cf `de ae `af `bf ad `bc `bd ce `cf `de ae `af `bf ad `bc `bd ce `cf `de ae `af `bf ad `bc `bd '.

Using (3.1), we can thus finally determine the limit of Pn " P0M n " pPRp0q PC p0q PT p0qqM n . thm 3.2. Suppose that |x1| " 1, x1 ‰ 1, and |x2| ‰ 1. Then the evolutionary model is not convergent. More precisely, we have:

• PRp2nq " pa 2 `2ab `ac ´2a `b2 `be ´2b `1qPRp0q `p´a 2 ´ab ´ac ´ad `2a `bf qPC p0q p´ab `ad ´b2 ´be ´bf `2bqPT p0q,

• PRp2n `1q " p1 ´a ´bqPRp0q `aPC p0q `bPT p0q,

• PC p2nq " p´ac ´bc ´c2 ´cd `2c `deqPRp0q `pac `c2 `2cd ´2c `d2 `df ´2d `1qPC p0q pbc ´cd ´d2 ´de ´df `2dqPT p0q,

• PC p2n `1q " cPRp0q `p1 ´c ´dqPC p0q `dPT p0q,

• PT p2nq " p´ae ´be `cf ´e2 ´ef `2eqPRp0q `pae ´cf ´df ´ef ´f 2 `2f qPC p0q `pbe df `e2 `2ef ´2e `f 2 ´2f `1qPT p0q,

• PT p2n `1q " ePRp0q `f PC p0q `p1 ´e ´f qPT p0q, Proof. Suppose that |x1| " 1 and |x2| ‰ 1. Then x1, x2 P R, and so x1 " 1 or x1 " ´1. The first case has yet been regarded.

If x1 " ´1, then ´s `2 ´?∆ " ´2 (due to (3.2)). So s " 4 ´?∆, and so s 2 ´4p " 4 ´4s `s2 . Consequently, p " s ´1. But x1x2 " 1 ´s `p, so x1x2 " 0, which leads to x2 " 0. Using (3.5), we can thus conclude that an " 1 ´p´1q n ´1 ´2 " 1 `p´1q n 2 . So a2n " 1 and a2n`1 " 0. Similarly, b2n " 0 and b2n`1 " 1, and finally cn " 0, @n P N. These values for an, bn, and cn lead to the following values for M n :

• PC pnq ÝÑ 4 s 2 pae `af `bf q,

• PT pnq ÝÑ 4 s 2 pad `bc `bdq.

Proof. In that case ∆ " 0, meaning that (3.

Competing Interests

The authors declare that no competing interests exist.

Peer-review history:

The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) www.sciencedomain.org/review-history.php?iid=728&id=6&aid=6913


Sandra Zimmer
email: sandra.zimmer@metz.ensam.fr

Laurent Langlois

Julien Laye
email: j.laye@institutdesoudure.com

Jean-Claude Goussain

Patrick Martin

Régis Bigot

Methodology for qualifying a Friction Stir Welding equipment

Keywords: Friction Stir Welding, Industrialization, Friction Stir Welding Equipment

The objective of this research work is the industrialization of the friction stir welding process in order to provide tools to industrials to select and qualify a machine for their FSW applications. This paper presents a methodology to determine the Friction Stir Welding equipment adequate to an application. The adequate equipment can be every machine that can perform friction stir welds. This paper presents a short review, based on literature survey, of the existing friction stir welding equipments. Then, the methodology developed is presented. It is based on the studying of the interactions between the tool and the workpiece.

Introduction

Friction Stir Welding is an innovative welding process. Its main advantages regarding other fusion welding processes are higher weld mechanical properties, low distortion and its automatization and repeatability possibilities. As it is a solid state welding process [1], it gives him the availability to weld almost all type of aluminium alloys, even the one classified as non-weldable because of cracking and porosity in the fusion zone [2]. Furthermore, it meets the need from industrials to join similar or dissimilar aluminium parts together or with other metals like cupper or steel. This should be convincing arguments for industrials to use it.

However, high forces are generated during the welding operation [START_REF] Smith | Robotic Friction Stir Welding using a Standard Industrial Robot 2 nd International FSW Symposium[END_REF] [START_REF]EREINER Prozessmodelle zur statischen Auslegung von Anlagen für das Friction Stir Welding ISBN[END_REF], described generally to be the most disadvantage of this welding process. The welding equipment must be rigid enough to satisfy them [START_REF] Strombeck | SANTOS Robotic Friction Stir Welding -Tool Technology and Applications 2 nd International FSW Symposium[END_REF]. Furthermore, the machine should be able to weld complex joint geometry to offer a wide range of applications. So, to produce industrial weld, special dedicated machine were developed. These kinds of equipments, as not standard, are generally expensive. Therefore, the industrials are reluctant to adopt this new technology. So, if the production cost could be reduced this technology will certainly widespread in terms of applications. One way to reduce the production cost is to choose a standard machine to perform FSW weld, like a robot, milling machine, etc. The machine can be every machine able to apply the requiring forces, tool kinematics with enough precisions to perform FSW welds.

The objective of this research work is the industrialization of the friction stir welding process in order to provide tools to industrials to select and qualify a machine for their FSW applications. The industrialization can be split into three aspects, as illustrated by the Figure 1, the analysis of the product, the process and the resources. These three entities are in interaction. The quality of mechanical parts depends on the expression of specifications (shapes, functions, dimensions, surface quality, and materials), the ability of shaping processes and resource capabilities. Therefore, to qualify a FSW equipment, the methodology focalizes on the study of the Process / Resources interaction. It leads to the determinations of the characteristics parameters in order to write down the technical requirements for the equipment. 2 The goal of the research work is to identify and to quantify the characteristics parameters to qualify a FSW equipment and the whole clamping device. Therefore a methodology will be developed. The base of the methodology is the definition of the characteristic parameters, for example the forces or torque, to qualify a FSW equipment. These characteristic parameters will be the basis of the establishment of the machine's technical specifications to choose or design the right equipment for a given application. This paper presents the methodology to determine the Friction Stir Welding equipment adequate to an application. The adequate equipment can be every machine that can perform friction stir welds. Therefore, this paper presents a short review, based on literature survey, of the existing friction stir welding equipments. Three types of architecture are analysed: milling machines, dedicated FSW machines and industrial robots including Tricepts ones. Then, the methodology developed is presented.

The existing friction stir equipment

Today, the machine commonly used to perform FSW, industrially or experimentally, are dedicated FSW machine, modified milling machine, serial and parallel kinematics robots. They all have their advantages and disadvantages but they all have to apply a high load on the tool, a spindle torque and permit the following of the seam. According to the seam geometry, a 2-or 3-dimensional machine will be required [START_REF]YAVUZ Function-oriented design of a friction stir welding robot[END_REF]. Furthermore, the workpiece geometry will also define the machine required workspace and the joint accessibility.

a. Modified milling machine

Milling machines were the first FSW machine. Furthermore, milling and FSW machine have the same general characteristics [START_REF] Ponte | LERTORA Low-cost transformation of conventional milling machine into a simple FSW work station Advanced Manufacturing and Technology[END_REF], spindle rotation, travel speed and CNC control process. They generally offer high stiffness and a good accuracy. They are widely used for machining operation in the industry. Therefore, a wide range of machines exist, from a 2 to 5 axes machines allowing 2 to 3 dimensional welds capabilities. They could apply until 20kN axial load and weld up to 15mm thick aluminium plates [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF]. The biggest disadvantage of most the milling machine is unavailability to propose a force controlled operation [START_REF]EREINER Prozessmodelle zur statischen Auslegung von Anlagen für das Friction Stir Welding ISBN[END_REF]. But, new milling machines technologies can offer a force controlled process.

b. FSW dedicated machine

FSW dedicated machine allows a force control operation and generally the possibility to weld with the three tool technologies, conventional, retracting and bobbin tool. This system allows controlling the tool / workpiece kinematics and the processing parameters. Each machine is designed to weld, 2-or 3-dimensional and has a predefined workspace. Some of them have a clamping device incorporated. The main advantages of these machines are their high payload capacity due to high framework stiffness. They permit to weld all types of material and a large range of thickness. The machines enabling the welding of 3-dimensional joint generally are composed of a five axes machine based on a gantry and two rotational axes [START_REF] Soron | Towards Multidimensionality and Flexibility in Friction Stir Welding Using an Industrial Robot System Recent Advances in Friction Welding and Allied Processes[END_REF]. They generally offer large workspace. The disadvantages of this kind of equipment is its high investments cost and the accessibility of the tool, because there is often one way to achieve a desired location and orientation [START_REF] Soron | Towards Multidimensionality and Flexibility in Friction Stir Welding Using an Industrial Robot System Recent Advances in Friction Welding and Allied Processes[END_REF]. The dedicated FSW machines are generally assimilated to high investment cost and low productivity [START_REF] Smith | Robotic Friction Stir Welding using a Standard Industrial Robot 4 th FSW Symposium[END_REF]. Therefore, other machine types have been investigated, in order to allow the welding in 3-dimensional configuration, increase productivity and reduce the investment cost.

c. Parallel kinematics robot

FSW on Tricept 805 robot has been developed by GKSS research centre. This kind of machine is commonly used for high speed milling applications [START_REF] Zettler1 | Lohwasser A Study on Material Flow in FSW of AA[END_REF] Tricept robots are characterized by high stiffness, allowing the application of 45kN down force, and to assure good position accuracy [START_REF] Strombeck | SANTOS Robotic Friction Stir Welding -Tool Technology and Applications 2 nd International FSW Symposium[END_REF]. Tricept ensure the accuracy necessary on machining operations, accuracy probably not required for FSW application. However, it appeared to be a good equipment to perform FSW joint. Unfortunately, this kind of equipment possesses high investments costs and small workspace due to its design [START_REF] Soron | Towards Multidimensionality and Flexibility in Friction Stir Welding Using an Industrial Robot System Recent Advances in Friction Welding and Allied Processes[END_REF]. These two characteristics are major obstacle to widespread its industrial FSW utilization.

d. Serial kinematics robots

Modern industrial robots ensure flexibility, multidimensionality and lower investments cost than dedicated FSW machine [START_REF] Voellner | KELLENBERGER Robotic Friction Stir Welding AeroTech Congress & Exhibition[END_REF]. They also possess a highly developed user interface for rapid programming [START_REF] Soron | Towards Multidimensionality and Flexibility in Friction Stir Welding Using an Industrial Robot System Recent Advances in Friction Welding and Allied Processes[END_REF]. Their big disadvantage is their lack of force / load capacity due to their lack of stiffness [START_REF] Smith | Robotic Friction Stir Welding using a Standard Industrial Robot 2 nd International FSW Symposium[END_REF] and [START_REF] Voellner | KELLENBERGER Robotic Friction Stir Welding AeroTech Congress & Exhibition[END_REF]. Therefore, to assure the robot welding capabilities and provide him a wide range of applications, the robot must be a high payload robot. The welding difficulties and limited capacities of a low payload (< 500kg) robot have been underlined in previous research work [START_REF] Smith | Robotic Friction Stir Welding using a Standard Industrial Robot 2 nd International FSW Symposium[END_REF]. Furthermore, the robot must be force controlled during the operation, because it will allow to the robot to adjust for cases where they are insufficient down force caused by structural compliance [START_REF] Crawford | Strauss Modelling of friction stir welding for robotic implementation[END_REF]. The main disadvantage of this technology is the limited process forces applicable. They are limited by the maximum power of the robot engines and the maximum torques of the gears [START_REF] Voellner | KELLENBERGER Robotic Friction Stir Welding AeroTech Congress & Exhibition[END_REF]. Furthermore, their lack of stiffness involves the deviation of tool position according to the seam [START_REF] Voellner | KELLENBERGER Robotic Friction Stir Welding AeroTech Congress & Exhibition[END_REF] and [START_REF]VOELLNER 3-D FSW using a modified high payload robot 6 th International FSW Symposium[END_REF]. Despite these two disadvantages, the industrial robot seems to be a good FSW mean of production. The different publications on the robotic FSW, [START_REF] Smith | Robotic Friction Stir Welding using a Standard Industrial Robot 2 nd International FSW Symposium[END_REF] to [START_REF] Voellner | KELLENBERGER Robotic Friction Stir Welding AeroTech Congress & Exhibition[END_REF] show a real development improvement and a real potential for industrial applications. They appear to be a good compromise between complex weld geometry, range of applications, workspace and investments costs. Therefore, they could be the key of the FSW wide spreading. For that reason, the Institut de Soudure and the ENSAM Metz are working on the FSW robotization. They are working on the methodology to qualify a FSW equipment. It will be applied on the robotization of process in order to characterize the robot capacity to perform FSW welds and its applications.

Presentation of the methodology developed

The methodology proposed is aimed to write down the technical specifications for a FSW machine dedicated to one or several applications and to qualify it. To achieve this goal, the interactions between the tool / workpiece and tool / material are analysed. The analysis of the tool / workpiece interaction is a global approach. It leads to the determination of the position and orientation of the tool during welding, according to the welding surface. It also defines the workspace required. In the other way, the study of tool / material interaction is a more local approach. It describes the tool position and orientation, according to the welding surface. It also defines the tool kinematics and the mechanical load applied on the tool. The methodology can be illustrated by the Figure 2.

a. Analysis of the tool / workpiece interface -global approach

It is an analysis based on the tool and geometric aspect. It is a global approach. It leads to the definition of the tool position and orientation at each location of the tool path. The welding configuration generally imposes a specific inclination and orientation of the tool with respect to the welded surface [MISHRA, 2005]. The method is based on the study of the seam and the welding surface geometries. The first step is the definition of the seam line, L. L is defined as the line generated by the joining of two surfaces without any gape (Figure 3). The seam L can be described as being a succession of point M i .

M i i t r i n r i v
i i i i M v n t M R i r r r

Linked by a direction cosine matrix (DCM)

Figure 3: Definition of the seam local frame orientation

For each location M i of the welded path, a local orthonormal reference frame R Mi (M i , t i , n i , v i ) is defined. The vector t i is tangent to the seam line, L, and oriented according to the welding direction. The vector n i is normal to the plane tangent to the workpiece welded surfaces. The Figure 4 presents some example of the n i definition according to the workpiece geometry. The third vector v i is chosen in order to form an orthonormal reference frame.

M i M i

Anvil

Example case 1 Example case 2

Workpice 1 Workpice 2

Figure 4: Illustration of the definition of the vector n definition according to the workpiece geometry

So, the successions of points M i are defining the tool path and the successive definition of reference frames R Mi are qualifying the welding surface geometry. Thus, for each point M i the corresponding reference frame R Mi can be linked by direction cosine matrix to a fixed workpiece (or anvil) reference frame R P . The next step is to position and orientate the tool reference frame, R O (O, x, y, z), according to R Mi , depending on the welding configuration. The reference frames R O and R Mi can also be linked together by a direction cosine matrix. Therefore, it leads to the awareness of the tool orientation and inclination along the seam L. It also defines the position of the tool frame with respect to the workpiece frame, Figure 5.

M i i t r i n r i v r M i+k k i t + r k i n + r k i v + r Seam, L ) , , , ( i i i i M v n t M R i r r r ) , , , ( z y x O R O r r r y r x r

b. Study of the tool / Workpiece kinematics and the mechanical interaction

To perform the weld, to assure friction heat and to contain the stirred material under the shoulder, a vertical down force has to be applied by the tool on the surface. Beside this, the tool is animated by a travel velocity and a rotational movement. This combination of these three factors (in bold) assure the heat generation and plastic deformation leading to the generation of the weld. Thus, in order to perform FSW welds, a machine has to apply these three process parameters. The machine also has to ensure the position and inclination of the tool at each location leading to a good stirring and seam tracking. The Figure 6 presents the interaction between the tool and the workpiece during the welding phase. It also schematizes the mechanical load applied by the tool on the workpiece material. This overview leads to the right understanding of which characteristics should have a FSW machine. It points out that the qualification of a FSW machine can be split into two analyses, the relationship between the tool kinematics and the mechanical load generated on the tool and their magnitude. Thus, the mechanical load can be written: The torque components L x and L y presented in Figure 7 will not be taken into consideration and are assumed to be equal to zero.

Anvil O F z z r x r F tool Travel direction, v travel [mm/min] ω ω ω ω z C y L x L T z F y F x F F z y x O
F tool/workpiece and T tool/workpiece are the force and torque applied during welding on the tool and transmitted to the FSW equipment. Thus, they are important for the qualification of a FSW equipment. Futhermore, they remain not constant during the whole welding phase. Therefore it appears important to analyse the welding phases in detail. It leads to the next step of the methodology, the determination of the welding characteristics parameters throughout the analysis of the welding operation and the analyses of the force and torque evolutions.

Determining the process and machine characteristic parameters

After identifying the parameters relative to the FSW process to ensure the energy and the tool kinematics in order to achieve a sound weld, the next step of the methodology developed focussed on the tool -workpiece interaction during all the welding operation. The process output parameters are dependent on this interaction. The Figure 8 presents the input and output parameters related to a FSW operation. Therefore, to qualify a FSW equipment, an analysis of the forces and torques applied on the tool during welding is necessary. Generally the FSW operation is described by two main phases, the plunge and the welding phase [START_REF] Sheikhi | Herstellung und Bewertung der Umformbarkeit von reibrührgeschweissten Tailored Blanks aus Aluminiumlegierungen[END_REF] and [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF]. As the tool -workpiece interaction differs according to the welding phase, it is necessary to decompose the whole welding phase. So, the tool kinematics were analysed leading to the decomposition of the welding phase into six independent sub-phases: plunging, dwell time, acceleration, welding at constant speed, deceleration and pin retracting. The Figure 9 presents the tool kinematics and the corresponding phases during FSW. Each welding phase is investigated separately in order to find out the relevant parameters defining a FSW machine. The tool kinematics, the forces and torques have been monitored and analysed for each welding phases. During dwell time, the kinematics of the tool is concentrated on the rotation. By analysing the forces and torque curves it appears that this phase is not characteristic. So, for the remaining five phases, it is necessary to analyse the tool -workpiece mechanical interaction for each welding phases separately. All the experimentation results given in this document had been performed on the fully instrumented MTS ISTIR-10 machine at the Institut de Soudure.

a. The plunge:

This phase corresponds to the entrance of the tool into the "cold" material [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF]. The process parameters, for a position control operation, are:

• ω plunge [tr/min]
The rotational frequency,

• ∆z

[mm] The depth of penetration,

• v plunge [mm/min]
The penetration velocity • a plunge

[mm/min²] The acceleration of the pin penetration

According [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF] and [START_REF] Simar | A multiscale multiphysics investigation of aluminium friction stir welds from thermal modelling to mechanical properties through precipitation evolution and hardening[END_REF], the penetration phase is characterized by a brief peak of the vertical effort, when the shoulder comes into contact with the surface. The "peak" intensity should be depending on the penetration velocity [START_REF] Simar | A multiscale multiphysics investigation of aluminium friction stir welds from thermal modelling to mechanical properties through precipitation evolution and hardening[END_REF]. The Figure 10 shows the plunging effort and torque feedback obtained during experimentation. It may be noted that the maximal torque and force are obtained almost simultaneously, at the end of the penetration. In the last 25% remaining penetration, probably when the shoulder touches material displaced upwards by the pin penetration, the curves of the torque and force grow more rapidly until they reach a "peak". That maximum occurs at the end of the penetration, when the tool shoulder is completely in contact with the plate's surfaces.

Plunging force according to the penetration The same force F z and torque evolution has been observed for other thicknesses and processing parameters. The Figure 11 presents the evolution of the force feedback during penetration for different processing parameters. The evolution presents the same caracteristics. After the dwell time, the tool is animated by a movement along the seam. The tool velocity progresses from a velocity equal to zero to the one parameterized. It is the beginning of the weld generation. Therefore, it is also the beginning of the force control regulation. By assumption, when the parameterized speed is reached, this phase is finished.

Process parameters related to the tool kinematics are:

• ω

[tr/min] The rotational frequency

• v travel [mm/min] The travel velocity, v travel [0 ; v travel ] • a travel [mm/min²]
The acceleration Process parameters related to the application of the force control activation:

• v force [kN/min] Force rate, • a force [kN/min²] Acceleration rate, • F z [kN]
The down force The Figure 12 presents the evolution of the forces in the three directions, (O,x,y,z), and the spindle torque. The effect of the material heating during dwell time can be seen on the decrease of the force F z and the torque. As the tool is animated by a movement, the travel and the transversal forces are increasing. The torque decrease tendency seems not to be affected, certainly still benefiting from the dwell time material heating. And, the force F z managed to reach the parameterized force. The machine begins the regulation of the force to maintain it constant until the end of welding operation. The acceleration phase is followed by a phase where the welding speed, rotational and the down force F z are maintained constant.

The process parameters related to this phase are:

• ω welding

[tr/min] The rotational frequency

• v travel [mm/min] The travel velocity • F z [kN]
The down force

The Figure 13 shows the evolution of the forces and torque during the welding operation at constant speed. As the operation is force controlled, the down force F z remains constant. After increasing during the acceleration phase, the travel force and transverse force seems to stabilize. The torque begins again to rise, suggesting that the material in front of the tool is "colder". It seems to stabilize after a few seconds. Then, heat generation is certainly stabilized and provided by the tool kinematics. Welding at constant speed At the end of the welding seam, the tool decelerates to reach a speed equal to zero. During this phase, the down force is slacked of smoothly. This down force release leads to the beginning of the tool retracting.

F y F x C z , Torque F z v parametrized = v travel
Process parameters related to the tool velocity are:

• ω

[tr/min] The rotational frequency,

• d travel [mm/min²]
The deceleration, v travel [v travel ;0] Process parameters related to the application of the force control release:

• v force [kN/min] Force removal rate • a force [kN/min²]
Acceleration removal rate, Process parameters related to the pin removal:

• ∆z

[mm]

The clearance high • V

[mm/min] The removal velocity • A

[mm/min²] The acceleration When the tool starts its deceleration, the travel, transverse and torque seem to decrease (Figure 14). At the end of the deceleration, the down forced is released leading to the removal of the pin. The FSW operation can be considered to be ended as soon as the spindle rotational movement is stopped, when the pin is entirely out of the material. The force and torque evolutions described above are mostly the same according to the aluminium alloy and thickness. The Figure 15 presents the forces and torque evolution for two materials and two thicknesses. This tool / workpiece analysis throughout the forces and torque curve, for each welding phase, are essential in order to know how the FSW equipment will statically be loaded. Furthermore, to qualify a FSW machine it is important to have a good understanding of the whole process. This curve analysis had been completed by a numerical analysis on different material and thickness, to gives the maximal, middle and minimum forces and torque occurred during each welding phase. It permits to compare one phase to another, in order to determine which phase will be characteristic to qualify a FSW equipment.

Forces and torque during the welding of a 6000 series aluminum alloy of thickness 5mm

Forces and torque during the welding of a 7000 series aluminum alloy of thickness 20mm

Comparing the forces, torque and power for each phases

A study of the maximum, mean and middle value of the forces and torque applied on the tool during each welding phases was conducted for two aluminium alloys, 6000 and 7000 series, and several thicknesses, 3, 5, 7 and 20 mm. The plunge, in the 20mm thick plate, was performed with a drilling hole. All the welding experiments are done with the same tool geometry and the processing parameters chosen led to a sound weld. The charts presented beyond are the maximum values observed. The same tendency has been seen on the middle and mean values. As the machine will first be qualified on a static point of view, the analysis presented in this paper will draw a comparison of the maximum value measured. Thus, phase by phase, the forces F z , then F x , F y and C z will be analyzed.

a. Comparison of the force F z over the whole welding phase

For these experiments, the plunging phase is position controlled and the welding phase is force controlled. Therefore, the down force F z is not set up and his maximum value is depending on processing parameters. The study reveals that, for most cases, the plunging force is greater than the welding force, Figure 16. This seems to be consistent with the curve displayed on the literature [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF].

The maximal plunge force could be reduced by decreasing the plunge velocity [START_REF] Simar | A multiscale multiphysics investigation of aluminium friction stir welds from thermal modelling to mechanical properties through precipitation evolution and hardening[END_REF] or by adding a drilling hole before plunging [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF]. For the configuration, welding a 7000 aluminium series of 20mm, the plunging phase was done with the help of a drilled hole. In conclusion, the plunging force has to be considered for the qualification of the structure.

Down Force Fz [kN]

Maximal plunge force Welding force

b. Comparison of the force F x , F y over the whole welding phase

The forces F x and F y correspond respectively to the travel and transverse force during the acceleration, welding at constant speed and deceleration phases. It has been seen on mostly application that the forces according to the x-direction is predominant during welding at constant speed. There is a major difference between the welding phase at constant speed and the three other namely for the strongest thickness. In contrary, the maximal force according the y-direction occurs during the plunging phase, except when the plunging phase is helped by a drilled hole.

Forces according to the x-direction

Forces according to the y-direction

Force according to the x-direction [kN]

Plunging Acceleration Welding at constant speed Deceleration

Figure 17: Comparison of the force according to the x and y direction during the welding phase for different configuration

The Table 1 presents a comparison between the forces F x and F y with F z . The forces F x and F y are not insignificant against the forces F z . They represent 8 to 16% of the forces F z .

c. Comparison of the force C z over the whole welding phase

The Figure 18 presents the evolution of the spindle torque over the 4 welding phases. It appears that the torque is slightly higher on the plunging phase when the operation is done without a drilled hole.

d. Comparison of the power developed on the spindle over the whole welding phase

The total power can be defined by: (

] / [ ] [ ] / [ ] [ ] [
The numerical determination of the power reveals that the power related to the motion of tool, according to the direction x or z direction during plunging, is negligible against the total power. The Table 2 presents the ratio between the power necessary to ensure the tool motion and the total power. The power relative to the tool motion is negligible against the total power, whatever the configuration, i.e. it is independent of the thicknesses and material welded.

Plunge Acceleration Welding at constant speed Deceleration

Power ralated to the tool motion Total Power Furthermore, [SIMAR, 2004] demonstrated experimentally that the power relative to the tool motion was also negligible whatever the travel speed. Therefore, it can be assumed that the power is defined by:

welding z C Power ϖ × = (5)

e. Dynamic behaviour of the forces

To qualify a FSW equipment, it is also important to take into consideration the dynamic behaviour of the forces, because it will be send back to the machine and to the clamping device. By analysing the forces F x and F y an oscillation phenomena have been highlighted. A frequency analysis has then been performed through a Fast Fourier Transform analysis. Into the F x and F y oscillations, there are one low frequency and one frequency corresponding to the tool rotational speed. [START_REF]YAVUZ Function-oriented design of a friction stir welding robot[END_REF] mention a variable load on the tool due to a combination of the tool rotation and the two forces components F x and F y . This could perhaps explain the low frequency response visualized in the experiments. The Figure 19 presents the experimental results. The analysis was performed on a sample corresponding to the welding at constant speed phase. It seams important to take into account this phenomena because it could impact the machine and clamping device on a vibration point of view implying the tool deviation from the welding path. Therefore, the F x and F y responses have to be investigated more deeply in order to understand the origin of the phenomena.

f. Conclusion of this analyse

The study done on the forces and torques intend to determine statically a FSW equipment. It enables to concentrate on the phases which are characteristics. It appears that F y , F z and C z are predominant during the plunge and welding at constant phases. Furthermore, it appears that F x and F y are not insignificant according to F z and should be taken into account for an equipment static determination. This knowledge will permit to focus on these phases' process parameters in order to reduce the forces and torque applied on the tool. Thus, it will enable the FSW on machine with load / rigidity weaknesses. On the other hand, another way to reduce the process forces is to work on the tool geometry [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF].

Application

The application presents the welding of two dissimilar aluminium alloys, 2000 with 6000 series. The Figure 20 presents the geometry of the two workpieces welded. The thickness of the two parts is the same. It is assumed that the workpiece surfaces form a perfect plane. The operation will be performed with a flat shoulder tilted at 0 degree angle. The 2000 aluminium alloy was placed on the advancing side. For this application, it didn't seem necessary to move the tool into one workpiece. With the assumption that the welded surfaces form a perfect plane and no gap remains after positioning the two workpieces, it could be assumed that for every location of the seam L, the point M i is congruent with the point O. So, by deduction, for this configuration, the reference frames R Mi and R O are superimposed. Therefore, the tool workpiece configuration required for this application, can be described on Figure 21.

a. Analysis of the tool / workpiece interface -global approach

The tool / workpiece global approach leads to the definition of the seam L. It also defines the orientation of the reference frame R Mi at each location M i of the seam. The Figure 22 presents the seam L. For some points M i on L, the reference frame R Mi is defined. As the welding path is a circle, the vector t i will be defined as being tangent to the circle oriented according to the welding direction. The orientation of the reference frame R Mi is changing among the welding path. The orientation can be defined with the rotation angle γ. Therefore, the location and orientation, according to R P, of the reference frame R Mi can be defined as:

Welding direction

M i i t r i v r
The direction cosine matrix linking R Mi to R p is: Thus, the position and orientation of the tool, according to the reference frame R P can be defined with the equations ( 6), [START_REF] Ponte | LERTORA Low-cost transformation of conventional milling machine into a simple FSW work station Advanced Manufacturing and Technology[END_REF], [START_REF] Prof | Dipl.-Ing Loucas PAPADAKIS Friction Stir Welding with modern milling machines. Requirements, Approach and Application 5 th International FSW Symposium[END_REF] and [START_REF] Soron | Towards Multidimensionality and Flexibility in Friction Stir Welding Using an Industrial Robot System Recent Advances in Friction Welding and Allied Processes[END_REF]. The required machine workspace for this application will be defined by the size of the circle diameter.

          - - - =

b. Study of the tool kinematics and the mechanical interaction

The next step is to determine the tool kinematics and the mechanical interaction between the tool and the workpiece material. From a kinematics point of view, the geometry welded is a planar circular joint, requiring a two dimensional tool motion to fellow the seam. Thanks to the direction cosine matrix linking R Mi to R P , the travel speed can be written according to the fixed workpiece reference frame:

p travel p travel P workpiece tool y v x v v r r r ⋅ × + ⋅ × - = γ γ cos sin , / (10)
So, according to [START_REF]YAVUZ Function-oriented design of a friction stir welding robot[END_REF] to perform this FSW weld, 4 degrees of freedom are required. These degrees of freedom can be given to the FSW equipment or to the workpiece holding fixture. By assumption, the holding fixture is fixed and all the degrees of freedom are possessed by the FSW equipment. Thus, to perform this weld, a 3-axis milling machine or a serial and parallel kinematics or a 3-axis dedicated FSW machine could perform the weld. If the tool had a concave shoulder, the tool would be tilted by a certain angle. To weld this application with a tilted tool, a supplementary axis would be needed. Therefore, a 4-axis machine will be required.

The processing parameters, F z , ω and v travel are chosen in order to produce a sound weld. To determine these parameters, straight butt welds were performed with the same tool, materials and thicknesses. The processing parameters used to perform the circular weld are the ones determined by the welding of the straight butt welds. The experiments show that the processing parameters determined on a straight weld can be applied to a circular weld in our case. The welds obtained were sound welds, for both applications. During these tests, the forces in the three directions and the torque were measured on the ISTIR 10 gantry machine at the Institut de Soudure. The welding equipment will have to apply the forces and torque required to perform FSW along the circular welding path. Therefore, at each location of the point M i , the mechanical interaction between the tool and the material is known for each welding phases. The Figure 25 presents the maximal forces and torque according to the tool location on the circle. By comparing the axial and transverse forces occurring during straight butt welding and circular welding, major differences has been seen. The maximal values monitored could be different until 50% depending on the welding phase. The Table 3 The deference measured could be explained by: • the different clamping devices,

• the workpiece positioning, some gap could remain between the workpiece due to the circular geometry of the seam, • the different tool / workpiece kinematics, • the tool seam deviation into one workpiece, due to seam tracking defect or machine deformation due to the tool / workpiece mechanical interaction.

The origin of these differences and a more detailed forces curve analysis are part of the future research work. Therefore, it is important to take into consideration these differences when qualifying a FSW equipment by for example applying a safety factor when using the axial and transverse forces monitored during straight butt welding. These forces and torque are used to determine the solicitation on the selected FSW equipment. The forces and torque are determined at each machine axis. After choosing one machine, the calculated torque and forces can be compared to the maximal admissible motor torque and forces at each machine axis. If the calculated torque and forces are lower than the admissible one at each machine axis, then the chosen machine can be assumed to be statically qualified for this application. The realization of the application on the chosen machine will confirm that the machine is dynamically able to perform this FSW application. The required spindle power is also defined and can be compared to the maximal machine spindle power. The knowledge of the forces and torque applied on the workpieces also permit to qualify the anvil and the whole clamping device.

Angle

The static determination of the welding device has for goal to make a pre-selection among the one existing. It also permits to pre dimension the equipment, motors, gears, during the FSW machine design.

Conclusion and perspectives

The presented work is the foundation of the methodology developed on this research work. The first step of this methodology was the study of the whole welding phase. The analysis was decomposed into a global aspect defining the seam geometry aspect and a local aspect in order to determine the tool workpiece interaction. The welding phase were decomposed it into six characteristic sub-phases. For each sub-phase, the welding processing parameters were defined and the resulting forces and torque were analysed. It provides a good understanding of the forces and torque evolution during a Friction Stir Welding operation. These knowledge are important to qualify a FSW equipment. The forces and torque applied on the tool during welding will be the input data to determine the operation feasibility on a specific machine. This methodology was applied to the circular welding of dissimilar material.

The future work will be concentrated on the welding capacities of different machines. With the forces and torque as input data, the forces and torque will be calculated on each machine's axis and compared to the admissible ones. Then, the most favourable position of the workpiece inside the machine's workspace, i.e. the one allowing the lowest motor torque and the best tool accessibility will be searched. On the other hand, a work will be done on the processing parameters, through the determination of the process windows, to reduce the forces and torque applied on the tool workpieces interface in order to facilitate the use of a specific machine.

Figure 1 :

 1 e r a c t i o n I n t e r a c t i o n FSW Analyse of the machines capabilities to perform FSW welds Analyse of the tool / Workpiece interface Illustration of the industrialization method

Figure 2 :

 2 Figure 2: Description of the global methodology to establih the bases of the qualification of a FSW machine.

  Perpendicular to the workpiece 2 surface n r Perpendicular to the vector t and to the plane tangent to the welded surface

Figure 5 :

 5 Figure 5: Definition to the tool position and orientation and each location of the welding path

Figure 6 :

 6 Figure 6: Schematization of the mechanical interaction between the tool and the workpieceThe tool travel velocity, named v travel , is defined as being the tool velocity on point O according to the welded workpiece.

Figure 7 :

 7 Figure 7: Mechanical load generated during the welding phase of a FSW operation

ForcesFigure 8 :

 8 Figure 8: Presentation of the input and parameters related to a FSW operation

Figure 9 :

 9 Figure 9: Presentation of the tool kinematics during the welding phase

Figure 10 :

 10 Figure 10: Evolution of the plunging force and torque feedback according to the plunge depth

Figure 11 :

 11 Figure 11: Evolution of the force Fz during the penetration, for the same plunging depth and different processing parameters b. Acceleration phase

Figure 12 :

 12 Figure 12: Forces and torque according to the time during the acceleration phase

Figure 13 :

 13 Figure 13: Forces and torque according to the time during the welding phase at constant speed d. Deceleration and tool material retracting

Figure 14 :

 14 Figure 14: Forces and torque according to the time during at the end of the welding operation

Figure 15 :

 15 Figure 15: Forces and torque during the welding operation according to the time for different aluminium alloys and thicknesses

Figure 16 :

 16 Figure 16: Comparison between the maximal plunging force and the applied welding force

Figure 18 :

 18 Figure 18 : Comparison of the maximal torque feedback on the different welding phases

Table 2 :

 2 Comparison of the power relative to the tool travel motion and the total power

Figure 19 :

 19 Figure 19: Analysis of the Fx dynamic response by comparing the forces measured and the FFT results

Figure 20 :

 20 Figure 20: Presentation of the workpiece's geometries

=Figure 21 :

 21 Figure 21: Schematization of the tool position and orientation according to the local reference frame RMi

Figure 22 :

 22 Figure 22: Illustration of the definition of the seam L L L L and the orientation of the reference frame R Mi at different location

AFigure 23 :

 23 Figure 23: Definition of the orientation of the reference frame R Mi according to R P

Figure 24 :

 24 Figure 24: Configuration to determine the welding parameters

Figure 25 :

 25 Figure 25: Presentation of the forces and torque according to tool location on the welding path

Example: butt welding of two plates

  r

																	Orientation of the local reference frame
																				i n + r	k
	Worpiece 1 Seam, L		y r	p				P		M i p z r	x r	p	Seam, L	i r v +	k	M i+k	i t + r	k
																	Surface formed by joining two pieces
																		workpiece interface
													Worpiece 2	Straight line tangent to the
																	seam at point M i
	Workpiece reference frame:	R	P	(	P	,	x r	p	,	y r	p	,	z r	p	)		Workpiece local frame on the seam at point M i :		(	,	,	,	)

Table 1 : Comparison between the forces in the plane

 1

		Average value for every configuration
		Plunge	Acceleration	Welding at constant speed	Deceleration
	Fx/Fz*100 [%]	9	10	16	10
	Fy/Fz*100 [%]	11	11	12	8

(O,X,Y) with the down force Fz

Fast Fourier Transform on the Fx Force

		1.2 1.2
		Fx
		1 1
		0.8 0.8
	Foce [kN]	0.6 0.6
		0.4 0.4
		Corresponding FFT
		0.2 0.2
		0 0
		58 58.2 58.4 58.6 58.8 59 59.2 59.4 59.6 59.8 60 60.2 60.4 60.6 60.8 61 61.2 61.4 61.6 61.8 62 58 58.2 58.4 58.6 58.8 59 59.2 59.4 59.6 59.8 60 60.2 60.4 60.6 60.8 61 61.2 61.4 61.6 61.8 62
		Time [sec] Time [sec]

Foce [kN] Fx Corresponding FFT Zoom on the Fx oscillation phenomena

		0.1
		0.09
		0.08
		0.07									Frequency corresponding
											to the tool rotational
	Amplitude	0.04 0.05 0.06									speed
		0.03
		0.02
		0.01
		0
		0	1	2	3	4	5	6	7	8	9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
											Frequency [hz]

  presents the results obtained

						Acceleration phase Acceleration phase		Welding at constant speed phase Welding at constant speed phase
					Fy [kN] Fy [kN]	Fx [kN] Fx [kN]	Cz [Nm] Cz [Nm]	Fy [kN] Fy [kN]	Fx [kN] Fx [kN]	Cz [Nm] Cz [Nm]
	Straight butt welding Straight butt welding	4.50 4.50	2.70 2.70	14.20 14.20	4.30 4.30	2.90 2.90	19.60 19.60
	Circular butt welding Circular butt welding	2.30 2.30	2.70 2.70	14.00 14.00	4.00 4.00	3.60 3.60	18.40 18.40
	Straight Circular	butt butt	welding welding	[%]	51.20 51.20	100.00 100.00	98.60 98.60	93.10 93.10	124.20 124.20	93.90 93.90

Table 3 : Comparison of the forces and torque measured during circular and straight butt welding. The given values are mean values measured on different successful trials realized with the same processing parameters

 3

th International Symposium on Friction Stir Welding -Awaji Island, Japan, 20-22 nd May 2008 -

Acknowledgment

The authors would like to thanks the Region Lorraine and the Moselle department to finance this research project.