Keywords: Relative Age Effect, Soccer, Dropout -2 -

The aim of this study was to examine the birthdates distribution of French male soccer players (n = 363,590) who dropped out during or after the 2006-2007 season, to test whether the Relative Age Effect (RAE) is linked to the dropout behaviour. An over-representation of players born late in the competitive year was found among dropouts, from the U9 to the U18 age categories, whereas their counterparts born earlier in the year were under-represented.

The distortion of the distribution was particularly important for the extreme quarters of the competitive year, and in the U13 and U15 categories. This biased distribution was however not confirmed regarding the U7 category. Finally, regarding the adult category, an unbiased distribution of dropouts was found as hypothesized. Practical implications to avoid the negative consequences of RAE are advanced.

Introduction

In order to equalise competition, sport organizations set up systems of age categories, based on the birthdates of young participants. Most of the time, those categories comprise 2 consecutive years. Yet, even if the primary purpose of such systems is to enhance the chances of success for every child, this way of grouping young players is likely to generate important differences in terms of relative age. Indeed, two children belonging to the same category may have up to 23 months of differences in age. This can lead to significant differences concerning physical [START_REF] Tanner | Clinical longitudinal standards for height, weight, height velocity and weight velocity and the stages of puberty[END_REF] and cognitive development [START_REF] Bisanz | Effects of age and schooling on the acquisition of elementary quantitative skills[END_REF], particularly around the time of puberty. In their study on young French basketball players, Delorme and Raspaud (2009a) observed important discrepancies in height between players from the same age category, depending on their quarter (i.e., three consecutive months) of birth. The largest mean differences approximated 10 centimetres among girls from the 11-12 years old category, and almost reached 12 centimetres for the male 13-14 years old category. Thus, given these differences in relative age, children born early in the competitive year possess developmental advantages (i.e., in height, body mass, and strength) which will probably have a positive impact on their potential as perceived by external experts in the field [START_REF] Helsen | The relative age effect in youth soccer across Europe[END_REF]. Those young participants are more frequently identified as gifted or promising, and are more easily recruited in training centres and/or national groups, which often represents the first step of a professional career. This early exposure to high level coaching may accelerate development leading to more elaborated technical and strategic competencies [START_REF] Williams | Perceptual skill in soccer: Implications for talent identification and development[END_REF][START_REF] Ward | Perceptual and cognitive skill development in soccer: The multidimensional nature of expert performance[END_REF]. Such a selection strategy (i.e., favouring an early physical development which may disadvantage athletes born late in the competitive year) leads to an unbalanced distribution of birthdates among elite players, with an over-representation of players born at the beginning of the competitive year, and an under-representation of players born later. This phenomenon was initially observed by [START_REF] Barnsley | Hockey success and birthdate: The RAE[END_REF] and has been called the relative age effect (RAE). Given the potential financial gains for athletes in certain sports, the RAE is seen as discriminatory, because it significantly reduces the chances of progressing to the elite for young players born late in the competitive year [START_REF] Simmons | Season-of-birth bias in association football[END_REF][START_REF] Edgar | Season of birth distribution of elite tennis players[END_REF]; for an exhaustive review on the determinants and mechanisms of the RAE in sport, see [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF] and [START_REF] Cobley | Annual age-grouping and athlete development. A meta-analytical review of relative age effects in sport[END_REF]).

Together with ice hockey, soccer has been to date the most studied sport with regard to this phenomenon. The RAE has been detected in professional championships of numerous countries from several continents: in the United Kingdom [START_REF] Dudink | Birth date and sporting success[END_REF], in Belgium [START_REF] Helsen | The influence of relative age on success and dropout in male soccer players[END_REF]), in Spain (González Aramendi, 2007), in France, in the Netherlands [START_REF] Verhulst | Seasonal birth distribution of West European soccer players: A possible explanation[END_REF], in Australia, in Brazil, in Germany and in Japan [START_REF] Musch | The relative age effect in soccer: Cross-cultural evidence for a systematic discrimination against children born late in the competition year[END_REF]. A similar, dissymmetrical distribution has also been observed among youth elite teams (e.g., [START_REF] Baxter-Jones | Growth and development of young athletes. Should competition levels be age related?[END_REF][START_REF] Brewer | Seasonal distribution amongst European soccer players[END_REF][START_REF] Helsen | The relative age effect in youth soccer across Europe[END_REF].

However, as noted by [START_REF] Wattie | Toward a unified understanding of relative age effects[END_REF], despite the growing body of studies, "the majority of relative age research in sport has sought to simply identify the existence (or non-existence) of RAEs in terms of who are selected for participation at a particular skill level". Yet, as underlined by [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF], RAE is not only suspected to generate discrimination in the selection process, but also to lead to dropout among less advantaged players (i.e., born at the end of the year). Indeed, because of the disadvantages resulting from relative age (i.e., less advanced physical development; [START_REF] Sherar | Do physical maturity and birth date predict talent in male youth ice hockey players[END_REF]Delorme & Raspaud, 2009a ), it is assumed that the young players born late in the competitive year experience more situations of failure and inferiority in their practice, and may be disadvantaged by a smaller time of play during official events. Indeed, [START_REF] Vaeyens | The relative age effect in soccer: A match-related perspective[END_REF] demonstrated that players born early in the competitive year received more opportunities to play than their late-born counterparts. Because of this less positive experience, it is likely that those young players will gauge lower feelings of competence in the activity and should ultimately stop to practice it. Past studies conducted on dropout from team sports actually revealed that a reduced time of play [START_REF] Guillet | Predicting persistence or withdrawal in female handballers with social exchange theory[END_REF] and a low perceived competence (Ommundsen & Vaglum, 1991a[START_REF] Ommundsen | The influence of low perceived soccer and social competence on later dropout from soccer: a prospective study of young boys[END_REF][START_REF] Guillet | Predicting persistence or withdrawal in female handballers with social exchange theory[END_REF] significantly account for a decreased psychological commitment and higher rates of dropout.

To our knowledge, only one study aimed at investigating whether RAE could account for dropout from the activity: [START_REF] Helsen | The influence of relative age on success and dropout in male soccer players[END_REF] were interested in RAE and its potential influence on success and persistence among Belgian soccer players.

They observed a regular distribution of birthdates in the 6-10 years old category but a biased distribution in the 12-16 years old category, and further inferred that "from 12 years on, there were a higher number of dropouts from those players born toward the end of the selection year" (p. 794). Nevertheless, no data were available concerning dropout during or at the end of the studied season. It is therefore not possible to know if this biased distribution from 12 years old on is the result of dropout, or a self-restriction before beginning the activity. Indeed, one can argue that because of the important differences in physical attributes, there are more beginners born early in the competition year compared to children born later, in this age category. The biased distribution could thus be due to the fact that new participants create an asymmetry which would not result from higher rates of dropout among players born late in the competitive year.

The purpose of this study is to examine the birthdates distribution of French male soccer players who ceased their participation in the middle or at the end of the 2006-2007 season, so as to test whether RAE is actually related to the dropout behaviour. We hypothesize that young players born late in the competitive year would be over-represented in the dropout sample. Conversely, because of their relative age and its associated advantages, the young players born early in the year should be under-represented in this sample.

Concerning the 'adults' category, we expect an unbiased distribution of the dropouts. Indeed, Baümler (quoted by [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF] found a negative correlation between RAE and age. The author suggests that the physical advantage of soccer players born in the first half in the competition year decreases gradually and corresponds to a shift towards players with better technical attributes, even though they may have suffered a disadvantage in terms of relative age when they started. Consequently, the dropout phenomenon should be independent of the RAE in the 'adults' category.

To sum up, the two following hypotheses are formulated:

1)

We expect a biased distribution of dropouts in young categories.

2)

We expect an unbiased distribution of dropouts in 'adults' category. collect the data, in order to avoid considering as dropout some players that took their licence while the new season had already begun.

Method

Among male players, the FSF distinguishes seven age categories: U7 (less than 7 years old), U9 (7 and 8 years old), U11 (9 and 10 years old), U13 (11 and 12 years old), U15 (13 and 14 years old), U18 (15, 16 and 17 years old) and adults (18 and more than 18 years old).

Data analysis

In the literature, the presence or absence of RAE is determined by examining whether a significant difference exists between the theoretical number of players born per month or quarter and the observed number of players. Hence, for each of the seven age categories distinguished by the FSF, the players' birthdates were classified into 4 quarters. As the cut-off date has changed in France [START_REF] Jullien | Influence de la date de naissance sur la carrière professionnelle des joueurs de football français [Influence of birth date on the career of French professional soccer players[END_REF], players born before 1982 were classified from Q1 (August-October) to Q4 (May-July), whereas those born in 1982 and after were classified from Q1 (January-March) to Q4 (October-December). Then, a chi square goodness-of-fit test was carried out to determine whether the observed distribution by quarter significantly differed from the theoretical distribution.

Usually, in the studies investigating RAE, the theoretical distribution is calculated based on the birthdates in the whole population of the country concerned, for the corresponding years, using weighted mean scores. This procedure implies an a priori postulate according to which the birthdates distribution among the whole population of licensed players in one sport is similar to the one of the general population. However, a recent study by [START_REF] Delorme | Relative Age Effect in Elite Sports: Methodological Bias or Real Discrimination[END_REF] indicates that when considering all male players of each age category of the FSF during the 2006-2007 season, the birthdates distribution was already biased, compared to the national data. This result suggests that the traditional method for examining the presence of RAE may not always be relevant and might introduce bias in the conclusions drawn up about this phenomenon. In this case, taking the national population as reference in order to calculate the theoretical distribution of birthdates may lead one to conclude to the absence of RAE in the dropout sample. As a consequence, in order to avoid this kind of bias, the theoretical distributions considered in the present study will be those from the corresponding population of licensed players, using weighted mean scores.

Results

Table It is also remarkable that among all youth categories, the U7, U9 and U11 represent the biggest categories in terms of participation, whereas the U15 is the smallest one. A significant distortion between the theoretical and observed distribution was revealed by the chi square test, regarding a majority of the youth categories, more precisely for the U9 (χ 2 = 141.71, d.f.= 3, P <.0001), U11 (χ 2 = 102.79, d.f.= 3, P <.0001), U13 (χ 2 = 188.30, d.f.= 3, P <.0001), U15 (χ 2 = 192.57, d.f.= 3, P <.0001) and U18 years (χ 2 = 45.42, d.f.= 3, P <.0001). In those five categories, the players born in the last two quarters of the competitive year were over-represented, whereas those born in the first two quarters of the year were under-represented, compared to the whole population of licensed players. The differences between theoretical and observed values were particularly high regarding the extreme quarters (i.e., the first and the last quarters of the competitive year). The asymmetry was particularly marked for U13 and U15 categories.

No significant difference between the theoretical and observed distribution appeared concerning the U7 (χ 2 = 4.81, d.f.= 3, P <.186) and adult categories (χ 2 = 1.51, d.f.= P <.680).

Discussion

In their review of the literature concerning the RAE in sport, [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF] underlined that recent work on sport dropout highlighted several psychological and sociological factors accounting for this phenomenon, but did not take into account the potential role played by RAE. Helsen and his colleagues (1998) sought to investigate this issue, but they failed to provide data regarding the distribution of dropout players' birthdates by quarter. The present study thus completed this line of research by examining whether the RAE was actually linked to the dropout behaviour.

Because of the differences in physical and cognitive development as a result of relative age, we hypothesized that a biased distribution should be observed among dropout players of youth categories from the FSF. Because they are less physically developed, and might not benefit from enough opportunities to play, especially around the time of puberty, the players born at the end of the competitive year should dropout from soccer in greater proportions, compared to their counterparts born at the beginning of it. In line with this assumption, an over-representation of players born in the last quarter was observed among dropout players, from the U9 to the U18 age categories, whereas the players born during the first two quarters were under-represented. The hypothesis was however not confirmed regarding the U7 category, where the birthdates distribution was similar to the one observed among all licensed players. Finally, regarding the 'adults' category, an unbiased distribution of dropout players' birthdates was found, as hypothesized.

In the U7 category, the absence of effect can be explained by two elements. Firstly the differences in terms of physical attributes remain modest. For instance, Delorme and Raspaud (2009a) showed that among young basketball players of 7 years old, the mean difference in height was only equal to 2.83 centimetres between players born in the first and the last quarters. The physical benefits associated with relative age thus seem limited for this age category. Next, it must be noted that the FSF does not organize any official competitions or championships for this age group. These young players only participate in recreational tournaments at the end of the season. According to [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF], competition is a necessary condition for RAE to emerge. Indeed, if there is low or no competition between players to belong to the team, every one of them may benefit from the same opportunities to play, whatever his/her physical development. As the authors notice, "the larger the pool of potential players, for a given sport in a given category, the strongest the resulting RAE should be" (p. 154). Thus the players from this category are less likely to be confronted to situations of failure or inferiority, and they may not see their time of play diminish at the benefit of other players. The low rate of dropout (8.1%) observed in this age group tends to support this explanation (see Table 1).

Regarding the other youth categories (i.e., from less than 9 to less than 18), the RAE hypothesis was confirmed. It should be noted that there is not yet an official championship for the U9 category. However, important physical disparities arise at this age among boys. For example, Delorme and Raspaud (2009a) reported a mean difference in height of 9.75 centimetres between less than 9 basketball players born in Q1 and Q4. This may imply a physical inferiority during the game and more frequent experiences of failure for the latter.

The dropout rate is almost doubled compared to the one observed in the younger category (see Table 1). For those players, as for those of older youth categories, the differences in physical attributes and the subsequent reduced opportunities to play, associated with lowered self-worth perceptions and perceived competence, are likely to lead to higher dropout rates (Ommundsen & Vaglum, 1991a[START_REF] Ommundsen | The influence of low perceived soccer and social competence on later dropout from soccer: a prospective study of young boys[END_REF][START_REF] Guillet | Predicting persistence or withdrawal in female handballers with social exchange theory[END_REF].

The categories showing the most important effects were the U13 and U15 groups (see Table 2). It is during this period that a identification process occurs and that selections are made by the FSF and decentralized organisations to orientate certain players toward elite groups [START_REF] Jullien | Influence de la date de naissance sur la carrière professionnelle des joueurs de football français [Influence of birth date on the career of French professional soccer players[END_REF]. In each age category, there is an increased competition, which is likely to enhance the RAE [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF] and might encourage dropout for the players born at the end of the year. It is also during this period that one can observe the greatest physical disparities since the puberty peak occurs around 14 years for French males [START_REF] Rochebrochard | Age at puberty of girls and boys in France: measurements from a survey on adolescent sexuality[END_REF]. Given this important concurrency associated to high differences in physical attributes, the RAE is thus maximised in these two age categories. Furthermore, there may also be conflicts between several activities practised during adolescence that are resolved when one sport is chosen against the others. The dropout of soccer found in this study may be the results of this change. In this vein, Delorme and Raspaud (2009b) found that in shooting sports, male membership holders are significantly over-represented in Q4 for the 15-17 years old category. This "reverse" RAE is the result of a greater participation in this activity by children born in Q4 compared to those born in Q1. These authors suggest that this phenomenon could be explained by the presence of "weak" players in the 13-14 years old category, who changed sports after failing in disciplines in which developed physical attributes are important for success. In the case of a transfer from one sport to another, children will continue to benefit of the multiple benefits of moderate but regular sport participation for social acceptance, psychological self-perceptions and health. The worse scenario is a definitive dropout from sport and/or physical activity.

It should also be pointed out that the most impressive dropout rate is the one observed for the U18 category (25.52%). This group comprises three consecutive years, instead of two for the younger groups. Finally, a distribution similar to the one observed among the whole population of licensed adult players was observed among adult dropout players. This result is consistent with past studies having shown that RAE gradually declines with age in the German professional championship (Baümler, quoted by [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF]. The physical advantages for players born in Q1 and Q2 decrease with time, and the diminution of the RAE can be viewed as the consequence of the prevalence of technical attributes as players grow up.

The RAE thus becomes inoperative and should not lead to dropout anymore. The high rate of dropout reported (23.49 %, see Table 2) masks an important variation according to age.

Indeed, this value steadily decreases from the beginning of this category (almost 41% of dropout for the 18 to 25 years) to its end (only 2.28% for the players aged 50 years and more).

The massive dropout at the beginning of adulthood is common to all sports and can be mainly explained by factors external to the activity itself. Indeed, field studies in sport psychology indicates that conflicts of interest arise with development, between sport and other domains, which negatively influence sport involvement (e.g., [START_REF] Boiché | Self-Determination of Contextual Motivation, Intercontextual Dynamics and Adolescents' Patterns of Sport Participation over Time[END_REF]. Those players have to assume educational, professional or familial responsibilities that might take them away from playgrounds. The potential influence of relative age on dropout should consequently be examined essentially among youth athletes.

Conclusions and Practical Implications

The results of the present study, together with those of [START_REF] Helsen | The influence of relative age on success and dropout in male soccer players[END_REF], confirm that the RAE can act as a sport dropout factor. This work is the first to bring empirical data on the biased distribution of birthdates of players that ceased their sport participation. The results can further be set in prospect with the study of [START_REF] Delorme | Relative Age Effect in Elite Sports: Methodological Bias or Real Discrimination[END_REF] in the more general realm of sport involvement. Indeed, even if dropout seems evenly distributed in the U7 category, those authors found a RAE for the whole population of licensed soccer players of this age. It means that a "self-restriction" process exits, by which children born late in the competitive year do not even start to practice this sport, when children born in the first part of the year are over-represented among beginners. The RAE thus show two potential paths of influence on involvement: on the one hand, children born at the end of the competitive year are dissuaded to engage in sport; on the other hand, those who do engage are more prone to dropout from it a few years later.

Because sport is aimed at promoting the development of every child, some authors have made suggestions so as to attenuate this effect and maintain motivation among all participants [START_REF] Musch | Unequal competition as an impediment to personal development: A review of the relative age effect in sport[END_REF], for example, building teams based on quarters of birthdates.

However, those suggestions sometimes seem difficult to apply, given the small number of players in small teams. In any case, coaches' awareness of this phenomenon should be increased. They have a central role in the sport system and should be willing and able to explain to children and adolescents that their physical deficits are only temporary, that the gap will ultimately be filled with time/growth, and that more controllable factors are important to succeed in sport. Even if coaches seem the most likely to take an active part against this

  Data collection At the end of the 2007-2008 season, the birthdates of the French male players (n = 363,590) licensed during the 2006-2007 season and that had not reiterated their licence during the following season were collected through the database of the French Soccer Federation (FSF). The identification of dropout players was facilitated by the fact the FSF attributes one and only one number to each individual. We waited until the end of the 2007-2008 season to

  1 presents the total rates of dropout for each age category among male soccer players during the 2006-2007 season. **** Insert Table 1 approximately here **** It is noteworthy that the dropout rate constantly increases with age among youth categories. Whereas there are only 8.11% of dropout cases for U7 category, the rate is superior to 25% among the U18 group. Among adults, the dropout rate reaches 23.49%, when the mean turnover percentage all groups confounded is equal to 19.85%.

Table 2

 2

presents the birthdates distribution of dropout players for each age category during the 2006-2007 season. **** Insert Table 2 approximately here ****

  Thus, a player born in December who progresses to this category can potentially be opposed to players born 35 months earlier. It is consequently in this group that relative age is the most important. However, the highest disparities in birthdates distribution are not observed in this age group. It is noteworthy that in this particular case, even some of the players born in Q1 or Q2 are susceptible to experience the disadvantages linked to relative age. As a consequence, they should drop out from soccer in similar proportions than their counterparts born later in the year. This may explain the reduced intensity of the phenomenon for U18 players.

phenomenon, their efficacy in doing so might also depends from their organisations' politics, and the way their discourse is heard and supported by parents of players. Note: ∆ is the difference between observed distribution and theoretical expected distribution.


Pierre-Antoine Adragna
email: pierre-antoine.adragna@univ-savoie.fr

Serge Samper
email: serge.samper@univ-savoie.fr

Maurice Pillet
email: maurice.pillet@univ-savoie.fr

Pierre-Antoine Adragna Is A Phd

A proposition of 3D inertial tolerancing to consider the statistical combination of the location and orientation deviations

Keywords: 3D tolerancing, inertial tolerancing, statistical tolerancing, form tolerancing, modal characterisation, statistical acceptance criterion

Tolerancing of assembly mechanisms is a major interest in the product life cycle. One can distinguish several models with growing complexity, from 1-dimensional (1D) to 3-dimensional (3D) (including form deviations), and two main tolerancing assumptions, the worst case and the statistical hypothesis. This paper presents an approach to 3D statistical tolerancing using a new acceptance criterion. Our approach is based on the 1D inertial acceptance criterion that is extended to 3D and form acceptance. The modal characterisation is used to describe the form deviation of a geometry as the combination of elementary deviations (location, orientation and form). The proposed 3D statistical tolerancing is applied on a simple mechanism with lever arm. It is also compared to the traditional worst-case tolerancing using a tolerance zone.

Introduction

Tolerancing of assembly systems is a large domain. Several tolerancing approaches are proposed with varying modelling complexity. The simplest modelling corresponds to the one-dimensional (1D) tolerancing, where only dimensional variations are considered as presented by [START_REF] Graves | Tolerance analysis tailored to your organization[END_REF]. Although this is the simplest tolerancing model, teams are still working on statistical methods with heterogenic and nonindependent components [START_REF] Anselmetti | Calcul statistique de chaînes de cotes avec distributions hétérogènes non indépendantes[END_REF]. [START_REF] Pillet | Inertial tolerancing in the case of assembled products[END_REF] proposes a statistical approach with a new criterion: the inertial tolerancing that will be used in this paper.

As this modelling may not be sufficient for complex mechanisms, three-dimensional (3D) modelling is proposed. As a nonexhaustive list of some approaches, one can distinguish the vector loop chain proposed by [START_REF] Chase | Tolerance analysis of 2D and 3D assemblies[END_REF], T-maps by [START_REF] Davidson | Geometric tolerances: a new application for line geometry and screws[END_REF], and the Small Displacement Torsor (SDT) used by [START_REF] Giordano | Clearance space in volumic dimensioning[END_REF], defined by [START_REF] Bourdet | The concept of small displacement torsor in metrology[END_REF]. This last 3D approach will be treated in this paper to express the tolerance domain associated with a tolerance zone. The tolerance values are calculated with a worst-case approach [START_REF] Giordano | Tolerance analysis and synthesis by means of clearance and deviation spaces[END_REF], and statistical analysis [START_REF] Germain | A new approach for three-dimensional statistical tolerancing[END_REF] is used.

Dealing with precision in the tolerancing of assembly systems, more complex models are proposed considering the form deviations of parts. Davidson's team proposes a model in the T-maps space to study the influence of form deviation [START_REF] Ameta | Influence of form frequency distribution for a 1D clearance which is generated from tolerance-maps[END_REF]. [START_REF] Radouani | Contribution à la Validation du Modèle des Chaines de Cotes -Etude Expérimentale du Comportement de la Liaison Plan sur Plan[END_REF] present an experimental study of the positioning regarding form deviation. [START_REF] Neville | The modeling and analysis of a butting assembly in the presence of workpiece surface roughness and part dimensional error[END_REF] proposes an algorithm to determine the positioning deviation of a butting assembly regarding the form deviation and roughness. Samper's team proposes an approach to determine the positioning error due to the form deviations of shapes for a given positioning force and auxiliary positioning surfaces (Adragna et al., 2007a).

The aim of this paper is to present an original 3D statistical tolerancing based on the fusion of the modal characterisation of form deviation and the inertial acceptance criterion of the 1D inertial tolerancing. This paper is a first step that only considers location and orientation deviations for the assembly of components. The article is decomposed into two main parts: a theoretical part that presents the fusion of the inertial and the modal approaches, and a part on the application of the 3D inertial tolerancing by the use of the small displacement torsor without form deviations. The assembly mechanism is a stack-up of three components with a lever arm so that the functional requirement is off-centre from the contact surfaces of the parts.

Methods used and background

This part presents three approaches:

1 inertial tolerancing and the associated inertial acceptance criterion 2 the modal parameterisation of any shapes to describe form deviations 3 the fusion of both approaches into the 3D inertial criterion.

The 1D inertial tolerancing

This part introduces the inertial acceptance criterion and a 1D statistical tolerance allocation called inertial tolerancing.

The inertial acceptance criterion

Inertial tolerancing concerns 1D models of mechanisms. Based on the Taguchi loss function, [START_REF] Pillet | Inertial tolerancing in the case of assembled products[END_REF] defines the quadratic off-centring of a batch to its target, called the batch inertia:

()

2 22 IXT = =+ (1)
where corresponds to the batch off-centring to its target and is its standard deviation.

To compare the deviation of a batch to its inertial tolerance, Pillet defines the Cpi capability index:

00 22 . II Cpi I == + (2)
Some compare this new criterion to the Taguchi capability index Cpm [START_REF] Chan | A new measure of process capability: Cpm[END_REF]. The main advantage of the inertial acceptance criterion is the absence of use of a tolerance interval that usually corresponds to the acceptance limits of the parts dimension. There is then no ambiguity of acceptance of a batch included in the tolerance interval but not acceptable due to the Cpm capability index. The representation of inertial tolerance uses two relations:

1 a tolerance on the off-centring considering the batch standard deviation:

2 2 0 Max I Cpi = (3)
2 a tolerance on the standard deviation considering the mean batch off-centring:

2 2 0 . Max I Cpi = (4)
Figure 1 shows a theoretical batch. The off-centring is = 0.010 mm, the standard deviation is = 0.020 mm and the inertial tolerance is I = 0.035 mm with a capability index Cpi = 1. Both diagrams in Figure 1 characterise the same batch. The figure on the right is the initial representation of the inertial tolerance. Because the figure on the left is closer to the traditional representation of a batch in its tolerance interval, it can be more easily understood.

Figure 1 The inertial tolerance and its ( , ) graphical representation (see online version for colours)

The inertial tolerancing method

Inertial tolerancing is a statistical method of tolerance synthesis. The allocation strategy is similar to that of the traditional statistical tolerancing. Let us consider components with centred batches. The resultant assembly is then a centred batch. The functional requirement is defined by a tolerance interval that is supposed to contain six standard deviations of the resultant batch. As the component inertias in the case of centring are given by the batch standard deviations, the inertial tolerances of components under the assumption of independent variables are given by: 0

6. i R I n = (5)
where:

I i = the inertial tolerance of the i-th components R 0 = the tolerance interval of the functional requirement n = the number of components in the dimensions chain.

In the general case, the inertial tolerance allocation is given by: 0 22 .

6.

.

i i ii R I = (6)
where i corresponds to influential coefficients of the i-th component on the resultant assembly and i is the feasibility index of the i-th component that allows a nonuniform distribution of the tolerances.

Improvement of the approach

Although the allocation of tolerances is made by a statistical approach, the 1D inertial tolerancing does not offer the disadvantages of the traditional 1D statistical tolerancing discussed by [START_REF] Graves | Five ways statistical tolerancing can fail, and what to do about them[END_REF]. Moreover, [START_REF] Parlar | Specification limits, capability indices, and process centering in assembly manufacture[END_REF] Off-centring tolerance Standard deviation tolerance discusses the interests of using the Taguchi capability index Cpm for the tolerance interval, which is equivalent to the Cpi index for inertial tolerance. Some of our works (Adragna et al., 2006a;2007a) show that the inertial tolerancing allows guaranteeing the functional requirement given by a tolerance interval R 0 and a Cpk index Cpk FR , or a tolerance interval R 0 and a nonconformity rate NCR FR . A very simple relation allows defining the components Cpi capability indices to guarantee the functional resultant Cpk FR index:

2 9 FR n Cpi Cpk =+ ( 7
)
where n is the number of components in the dimensional chain. The link between the component Cpi index and the maximum limit of the functional requirement NCR FR is given by the resultant distribution law, chosen as Gaussian, under the consideration of off-centred distribution.

Theoretical application

Let us consider a 1D tolerancing problem of a three-part stackup. The functional requirement is the total dimension of the assembly. The functional tolerance is chosen as t = 0.2 mm and the maximum nonconformity rate is NCR FR = 500 parts per million (ppm). Under the assumption of Gaussian distribution of the resultant assembly, the Cpk FR index corresponding to NCR FR is Cpk FR = 1.10. Thus the component Cpi index guaranteeing FR is Cpi = 1.24, given by Equation ( 7). The histogram in Figure 2 shows the obtained NCR of the resultant assembly for components with only positive off-centring and acceptable Cpi capability indices. Although these assumptions are unrealistic and pessimistic, one can observe that the maximum NCR does not exceed the NCR FR .

The modal characterisation of form deviations

This part presents a method to characterise the deviations of measured shapes. It is then possible to identify rigid deviations such as location and orientation, and several form deviations.

Origin of modal characterisation

The description of form deviations is a large domain for which one can find lots of propositions for specific applications, such as the famous Fourier transform applied to roughness filtering, the Zernicke polynomials used for the qualification of lens deviation, the Discrete Cosinus Transform (DCT) proposed by [START_REF] Huang | Mode-based decomposition of part form error by discrete-cosine-tranform with implementation to assembly stamping system with compliant parts[END_REF], and a combination of the Chebyshev polynomials and Fourier series used by [START_REF] Summerhays | Optimizing discrete point sample patterns and measurement date analysis on internal cylindrical surfaces with systematic form deviations[END_REF] to characterise the form deviations of cylinders. Although these approaches are efficient for the creation of a deviation basis and the analysis of form, they are applied on some specific shapes (profile, discus, rectangle or cylinder).

Introduced by [START_REF] Samper | Form defects tolerancing by natural modes analysis[END_REF], the modal characterisation of shape deviations is a generic approach that is able to describe form deviations of any geometry. Its main advantage is its form defects basis for the characterisation of measured form deviations. Each mode of the basis is defined by the shapes of the modal vibration of the ideal geometry to analyse. For practicality, the ideal geometry is discretised with nodes corresponding to measured nodes. Deformed shapes basis are natural mode shapes of the Finite Element Model (FEM) of the nominal geometry. This approach allows obtaining a modal basis for any shape, such as the gap flush profile of an upper hood [START_REF] Adragna | Modal tolerancing -application to gap and flush analyses[END_REF] or its surface (Favrelière et al., 2007a), the external cylinder of a hydraulic spool (Adragna et al., 2006c) and a sphere (Favrelière et al., 2007b).

In our application, the surfaces are the contact surface B i and C i and the functional requirement surfaces A i , which are rectangular planes. Although the modal vibration resolution of a rectangular plane can be analytically expressed, our solution is given by a FEM solver. As a good practice, some nodes' degrees of freedom are fixed: the translation degrees of freedom along the plane axes defining membrane vibration the rotation of nodes around the plane's normal direction. This avoids useless modes (form deviations) for the qualification of form deviation expressed along the plane's normal direction. Thus, a natural modal basis is obtained composed of three rigid body modes, originally expressed by the combination of rigid translation and rotations, while the other modes are form deviation shapes.

Evolution of the modal characterisation

Metric modes

The first modification of the modal basis consists of specifying the modes' amplitude to the unit. This allows linking the modal coefficient to the exact amplitude of the mode. Detailed in a further part, the characterisation of a measured deviation shape gives a modal signature containing all modes' influences for this analysed form deviation. The modal coefficients correspond to the influence of each mode for the description of the shape, and their values are expressed in the unit of the measured form deviation (mm, µm or inch).

Rigid modes

The second modification of the natural modal basis consists of specifying the rigid modes. In our case, the first rigid mode, 1 , is defined as a location deviation that corresponds to identical translation of the nodes along the plane's normal direction. The second and third rigid modes, 2 and 3 , can be interpreted as rotation deviations around chosen axes that are symmetrical axes of the plane.

This parameterisation can be compared to and translated into the SDT parameterisation. Figure 3 shows two of the three rigid deviation modes of the modal basis and two different SDT data positioned on the plane. These models can be linked thanks to linear relations expressed as a matrix. This last matrix can be diagonal by changing the rigid modes' shapes.

Application of the method

Figure 3 shows some of the first modes of the basis. The first image presents the meshing of the plane and the location of two SDT data used in the previous part. Then, two of the three rigid modes are shown. They can be named translation and rotations around the x axis. The other modes correspond to form deviations. Mode no. 6 can be classified as domed shape, mode no. 4 can be called chip mode, and other modes such as mode no. 20 can be classified as undulation modes.

A measurement D is characterised in the modal basis B. Because the modal basis B is not an orthogonal and normal basis, the i coefficients are identified by the dual basis. The modal signature , vector of the modal coefficients i , is then computed by: 1 (( . ). ). .

t BBBD = (10)
As the deviation shape is characterised by the modal signature = [ … i … ], the form deviation can be recomposed and the efficiency of its characterisation can be evaluated. The recomposed, or reconstructed, shape R of the form deviation is given by:

.. R B = (11)
The reconstruction error is given by the form difference between the measured shape D and its reconstructed shape R. One can then compute the scalar residue r given by the norm of the reconstruction error . It is even possible to define r(m), the scalar residue of the modal characterisation with m modes. One then defines a relative scalar (m,n) that is the ratio of the r(m) scalar residue over r(n). This finally allows comparing the efficiency of one mode or a lot of modes in the whole characterisation.

2.3 3D inertial tolerancing: fusion of the modal characterisation and the inertial criterion 2.3.1 The 3D inertial acceptance criterion Presented in Adragna et al. (2007b), the inertial acceptance criterion is applied to the 3D and form characterisation method, which is the modal parameterisation. [START_REF] Pillet | Geometrical inertial tolerancing', 6ème congrès Pluridisciplinaire Qualité et Sûreté de Fonctionnement[END_REF] introduced the inertial criterion on geometrical deviations. They hence define the inertia of a measurement set corresponding to deviation points. In the case of n surfaces measured by k deviation points, the surface batch inertia is defined as: where X i,j is the measurement of the i-th point on the j-th shape. The second sum corresponds to the computation of the inertia of the i-th point defined by Equation (1). Hence, the original definition of the surface batch inertia corresponds to the quadratic mean of the surface point inertias:

2 1 1 . n j j I I n = = (13)

Fusion into the modal parameterisation

As the modal parameterisation is a linear relation (Equation 11), [START_REF] Saporta | Probabilités Analyse des Données et Statistiques[END_REF] gives two statistical relations:

.

R B ∝ ∝ = ( 14
)
where ∝ corresponds to the mean modal signature of a set of modal signatures i and ∝ R corresponds to the mean shape. This mean shape indicates the mean deviation of each point of the characterised surface batch:

.. t R B B = ( 15
)
where R is the covariance matrix of the surfaces points and is the covariance matrix of the modal coefficients. Then, the standard deviation form of the characterised surfaces is the square root of the R diagonal. In relation to these statistical characterisations of a form deviation batch, Favrelière et al. (2007b) gives graphical representations of these mean and standard deviation shapes.

The inertial criterion of a form deviation batch described by the modal parameterisation can then be expressed as follows:

2 1 1 (( . ) ( . . )) n j I BdiagBB n ∝ = =+ ( 16
)
where k is the number of measured points on the surfaces and is also equal to the length of the mean shape B. ∝ . The diagonal of the recomposed shapes covariance matrix B. .B is the variance vector of length n. The advantages of using the modal parameterisation to compute the surface batch inertia is the reduction of the parameter number from k points to m modal coefficients with m<k, and the knowledge of the form deviation (chip, domed, etc.).

Detection of a disadvantage and a solution

Highlighting of a problem

As the surface inertia is the quadratic mean of point inertias, some configurations may cause problems. Let us consider a 3D model based on the location and orientation deviation modes applied to the theoretical example presented in Figure 2. This problem can also be treated as four 1D tolerancing problems at each part's border. Then, the 3D FR is guaranteed if all 1D FR are guaranteed.

To underline the disadvantage of the initial geometrical inertia, let us consider a case where both left 1D dimension chains are perfect (centred and not dispersed) but the right dimension chains have variations. In this theoretical case, the surface inertia of the lower part I is defined by the quadratic mean of both front and rear left inertias, I lf and I lr , and both front and rear right inertias, I rf and I rr considered as equally deviated:

22 1 (0 0 ) . 4 2 r rf rr I III =+++= (17)
Thus, the right corner variations are larger than the surface inertial tolerance. In this condition, the right dimensional chains do not guarantee its FR, thus the assembly FR is not guaranteed.

Solution: the 3D adjusted inertial criterion

To solve this problem, and in the assumption of no form deviation (only location and rotation deviations), the adjusted inertial criterion is proposed (Adragna et al., 2007b). The aim is to consider only the worst points inertia in order to avoid the compensation detailed in the previous part. The adjusted inertia is then defined as:

().

adj j I MaxI = (18)
This new definition applied on the 3D problem with modal parameters can be written as:

2222 123123 ()(). adj I =+++++ (19)
With the SDT datum centred on the O point, a similar definition can be found. The maximum deviation is given by: ... 22

z x z yx Max l l TRR =++ ( 20
)
The inertia is then given by: The definitions Equation ( 19) and Equation ( 21) are equivalent and the transition matrix of Equation ( 8). One can also note the assumption of an independent parameter due to the absence of the covariance. The covariance is not considered for now in these two relations, but is implicitly taken into account in the relation ( 18). In order to use the relations Equation ( 19) and ( 21) in a valid context, let us consider the assumption of independent parameters. In this case of 3D parameterisation, it is possible to represent the geometrical inertial deviation. We chose to represent the inertial deviation and not the inertial tolerance as for the 1D representation in Figure 1. Two kinds of inertial deviations are identified:

1 the mean deviations defining a triangular domain 2 the standard deviations defining an elliptic domain. One can note from the first image of the batch shapes that the rotation around the y-axis (mode 3) is the largest dispersion. This can be observed on the second representation of the batch in the modal domain. The largest direction of the ellipsoid is along the mode 3 axis. Finally, the last representation of the batch deviation shows that the maximum mean deviation is due to the translation (mode 1), completed with a little rotation (mode 3) that is observable in the blue tetrahedron. The standard deviation presented in the part of the green ellipsoid shows an important standard deviation on mode 3, and then on mode 2.

3D inertial tolerancing without form errors

This part proposes a statistical tolerancing approach based on the 3D adjusted inertia and the consideration of only rigid shapes (no form deviation). The case of application is the three-part stack-up with a lever arm, and FR is a location tolerance zone t = 0.2 mm. The statistical approach is compared to the worst-case tolerancing on a common model: the SDT, the data origins of which are placed on each face's centre. Comp. 1

Comp. 3 A1 C3 B2 B2 t1 B2 t2 A1 t1 A1 t2 C3 t1 Surf. A3 Surf. B1 Surf. C2
The qualitative tolerancing is arbitrarily chosen as follows:

a location tolerance zone applied on the three surfaces A, B and C. The size of the tolerance is t 1 . an orientation tolerance zone applied on the surfaces B and C. The size of this tolerance is t 2 . t 1 is twice as large as t 2 . The mechanism characteristics are L Bx = 80 mm, L Ax = 100 mm, L Az = 80 mm and d = 220 mm.

The quantitative tolerancing with a worst-case approach gives t 1 = 0.0229 mm and t 2 = 0.0114 mm. This result is found with the deviation domain method [START_REF] Giordano | Tolerance analysis and synthesis by means of clearance and deviation spaces[END_REF].

A proposition of 3D inertial tolerancing

Our proposition of inertial tolerancing consists of identifying dimensional chains for each component of the SDT. In the treated case, three dimensional chains are identified:

1 translation of each surface on the y axis 2 rotation of each surface around the x axis 3 rotation of each surface around the z axis.

These three dimensional chains give the maximum inertia on each axis of the SDT for each component. In order to have the same restricted tolerance distribution as the worst-case tolerance, it is chosen to not uniformly distribute the tolerance on the rotations' components. As with the worst case, the rotation tolerances of component 3 are twice as large as the rotation tolerances of components 1 and 2.

Allocation of the inertial tolerances

Inertial tolerancing of translations

The dimension chain of the translations on the z axis is:

123 . Max z z z TTT =++ (23)
Thus the inertial tolerancing relation with the assumption of uniform distribution of the tolerance is:

. 6. 3 Tzi t I = (24)
The components inertial tolerances are I Tzi = 0.0192 mm.

Inertial tolerancing of rotation around x

The dimension chain of the rotations around the x axis is:

133 .... 222 Ay Ay Ay Max x x x L LL RRR =++ (25)
The tolerance distribution uses the feasibility indices 1 = 2 = 1 and 3 = 2, thus:

2 . . 3. . i Rxi Ay i i t I L = (26)
The components' inertial tolerances are then I Rx1 = I Rx2 = 3.4.10 -4 rad and I Rx3 = 6.8.10 -4 rad.

Inertial tolerancing of rotation around z

The dimension chain of the rotations around the y axis is:

123 .... 222 Bx Bx Ax Max y y y L LL RdRdR =++++ (27)
The tolerance distribution uses the feasibility indices 1 = 2 = 1 and 3 = 2, thus:

22 22 13 . . 6. 2. . . 22 i Ryi Bx Ax t I LL d = ++ (28)
The components' inertial tolerances are then I Ry1 = I Ry2 = 8.4.10 -5 rad and I Ry3 = 1.7.10 -4 rad.

Identification of combination laws

As the SDT components do not have the same tolerance, a combination law has to be determined. The inertial tolerance of the translation components is the larger tolerance; it is thus chosen as the reference tolerance. The influence of the two rotation components is identified by the ratio of the inertial tolerances. It is also possible to analytically identify these combination coefficients. The combination ratio is numerically identified as follows:

Rx ≈ 56.6 and Rx ≈ 227.9 for components 1 and 2 Ry ≈ 28.3 and Ry ≈ 113.9 for component 3.

Comparison to the worst-case tolerances

This part presents a comparison of the dispersion of both tolerancing approaches in the case of centred distribution of the components' deviations. [START_REF] Germain | A new approach for three-dimensional statistical tolerancing[END_REF] shows that, in the case of a centred multinormal distribution of the components' deviations with six standard deviations within the tolerance interval on each SDT component, the filtering of parts out of the components' tolerance interval is not necessary due to the low resulting NCR on the resultant assembly. Since the comparison of a worst-case tolerance and a statistical tolerance would not be exact if different hypotheses are used, the same hypothesis is considered to compare the two tolerancing approaches.

Let us consider a multinormal distribution within the tolerance interval (six standard deviations within the tolerance zone for each SDT component). The parts distribution associated with the inertial tolerance has a similar repartition to the worst-case distribution, hence the comparison identifies the ratio of a homothetic transformation.

Figure 6 compares the multinormal distribution of components 1 and 2. The distribution is represented by a 3D ellipsoid representing +/-3 standard deviations. This representation allows an easier comparison; a set of points would have hidden the border of the worst-case tolerance domain. One can observe in this homothetic comparison that the inertial distribution is 63% larger than the worst case distribution. Figure 7 presents the comparison of the +/-3 standard deviation ellipsoids for component 3. In this case, the homothetic increase is 95% of the worst-case centred distribution.

The aim of tolerancing is to guarantee the final resultant assembly. In order to verify if the proposed tolerancing guarantees the respect of the FR, the characteristics (mean and covariance matrix) of resultant assembly are calculated as presented by [START_REF] Germain | A new approach for three-dimensional statistical tolerancing[END_REF]. Then 1 000 000 resultant assemblies are randomly drawn with a multinormal distribution and the resultant assembly covariance matrix. Figure 8 presents both the resultant assembly for the statistical assemblies based on the worst-case or inertial tolerances of the two previous figures. The dark points represent the assemblies' characteristics. The assemblies out of the FR (domain delimited by the blue lines) are included in the red circles. The NCR obtained for the statistical distributions based on the worst-case tolerancing is negligible; NCR WCT is a few ppm as presented by [START_REF] Germain | A new approach for three-dimensional statistical tolerancing[END_REF]. For the tolerances based on the inertial tolerancing, the NCR In. is around 3000 ppm = 3%. Although this resultant NCR In. is greater, it can be reduced by the use of the Cpi capability index. A summary of some assembly NCR In. in the case of centred components is presented in Table 1.

Worst case tolerance domain

Worst case components distribution

Inertial components distribution

Analysis of the 3D inertial tolerances

The first simulation, presented as the configuration 'Centred 1' in Table 1, is illustrated in Figure 8 and corresponds to centred components with the standard deviation distribution detailed in Part 3.1.2. The second simulation, 'Centred 2' in Table 1, is illustrated in Figure 9 and considers centred components with a random standard deviation distribution.

The third simulation, 'Off-centred' in Table 1, is illustrated in Figure 10 and considers randomly off-centred components with a random distribution of the standard deviation. In order to avoid compensations of the components' off-centring and to evaluate a kind of worst configuration, components' off-centring are all considered positive.

Table 1 gives the results for the assembly NCR in the case of centred components and the maximum reached NCR with the hypothesis of random distribution of the components' standard deviation or random distributions of the component's off-centring and standard deviation.

The first line, Centred 1, corresponds to the particular case where all components are centred and their dispersion is similar to the one associated with the worst-case tolerances, presented in Figures 678. The indicated results are expressed in ppm. The NCR is the mean result of 2000 simulations and its standard deviation, also in ppm, is indicated. One can note that the observed standard deviation of the results follows the prediction of [START_REF] Cvetko | New metrics for evaluating monte carlo tolerance analysis of assemblies[END_REF]

: 6 .(10 ) 1 NCR NCR NCR N = ( 29
)
where N is the number of drawings.

For the two other lines, only one value is indicated that corresponds to the maximum NCR value obtained during our simulations. As the worst NCR is not easily obtained, components' off-centring are limited to a third of their inertial tolerance. Without this condition, the worst NCR is not drawn. To illustrate our purpose, Figure 11 shows two simulations for the Cpi capability index but different hypotheses on the components' off-centring. The histogram on the left is the result of random drawing of components' off-centring within a third of their inertial tolerance, and the second histogram is a random drawing of the off-centring within their entire inertial tolerance. One can note that the maximum NCR obtained in the second histogram is lower than the first one.

The second line, Centred 2, corresponds to centred components with random dispersion as presented in Figure 9. Although the multinormal distribution has a random allocation of its components' standard deviation, the capability indices of each components' batch is set to the specified value.

Finally, the last line, Off-centred, corresponds to the assembly simulation of randomly off-centred components and randomly distributed standard deviations. The assembly considers only positive off-centring of each component as presented in Figure 10.

From this table, one can conclude that the Cpi capability index has a real effect on the maximum obtained NCR. The proposed 3D inertial criterion seems to be a good criterion for 3D statistical tolerancing. From this table, one can conclude that the Cpi capability index has a real effect on the maximum obtained NCR. The proposed 3D inertial criterion seems to be a good criterion for 3D statistical tolerancing.

Conclusion

This paper presents a new acceptance criterion for 3D statistical tolerancing. Derived from 1D statistical tolerancing, where it is efficient, the 3D inertia seems to be well adapted for the acceptance of rigid form deviations. The Cpi capability index has an influence on the resulting NCR. The results of this paper show why there is interest in this new acceptance criterion. Some of our next works will aim to evaluate the components' configuration that give the worst assembly NCR. Then, as for the 1D statistical tolerancing, it may be possible to give the Cpi index that exactly guarantees the FR. Another objective is the consideration of form deviations and their specification in order to guarantee the FR.

Figure 2

 2 Figure 2 Application and NCR results (see online version for colours)

Figure 3

 3 Figure 3 Some of the shapes of the modal basis (see online version for colours)

Figure 4

 4 Figure 4 presents a batch of form deviation, its representation in the domain of the first three modes and its inertial deviation in the same modal domain.

Figure 4 (

 4 Figure 4 (a) Batch of form deviations, (b) modal representation of the batch, (c) inertial deviation domains of the modal components (see online version for colours)

Figure 5

 5 Figure 5 (a) Batch of form deviations, (b) modal representation, (c) inertial deviation domains (see online version for colours)

Figure 6

 6 Figure 6 Comparison of the tolerances of components 1 and 2, inertial distribution is 63% larger than worst case distribution (see online version for colours)

Figure 7 Figure 8

 78 Figure 7 Comparison of the tolerances of component 3, inertial distribution is 95% larger than worst case distribution (see online version for colours)

Figure 9 Figure 9

 99 Figure 9 An assembly of centred components' batches, Cpi = 1.16, NCR = 633 ppm (see online version for colours)


Y Ledoux
email: yann.ledoux@univ-savoie.fr

H Favrelière

S Samper

OPTIMIZATION OF A CLASSICAL STAMPING PROGRESSION BY MODAL CORRECTION OF ANISOTROPY EARS

Keywords: numerical simulation, experimental design, modal defect description, discrete cosine transformation, optimization, stamping stage reduction, minimization of material use

This work is a development from the INETFORSMEP European project. It was proposed to realize a global optimization of a deep drawing industrial progression (made of several stages) for a cup manufacture. The objectives of the process were the thickness decrease and the geometrical parameters (especially the height). This paper completes this previous work in the aim of mastering the contour defect. From the optimal configuration, we are looking for minimizing the needed material and the number of forming operations. Our action is focused on the appearance of undesirable undulations (ears) located in the rim of the cups during forming, due to non-uniform crystallographic texture. These undulations can cause a significant amount of scrap, productivity loss and undesired cost during manufacture. In this paper, this phenomenon causes the use of five forming operations for the cup manufacture. The focus is to cut down from five to two forming stages by defining an optimal blank (size and shape). The advantage is to reduce the cost of the tool manufacturing and to minimize the needed material (by suppressing the part flange). The envisaged approach consists in defining a particular description of the ears' part by modal decomposition and then simulating several blank shapes and sizes generated by discrete cosine transformation (DCT). The use of a numerical simulation for the forming operation and the design of experiment technique allows to find mathematical links between the ears' formation and the DCT coefficients. An optimization is then possible by using mathematical links. This original approach leads to reduce the ears' amplitude by ten, with only fifteen numerical experiments. Moreover, we have downsized the number of forming stages from five to two with minimal material use.

Introduction

This work succeeds to a previous publication where it was realized an optimization of a stamping part formed through few stages [START_REF] Ledoux | Numerical optimisation of an industrial multi-pass stamping progression by using the design of experiment method[END_REF]. It was applied a global optimization of the whole process. The objectives were:

• to find the minimal blank dimension which leads to assure the flange cutting and the final height of the part. • to determine the best tool set configuration (tool radii, blank holder force, …) which leads to a minimal thickness decrease along the part.

The stamping progression was composed of 4 stages: cutting the initial blank, the forming operation, the flanging and the cutting operation. These later operations were necessary due to the anisotropy of the material. Because of this non-uniform crystallographic texture, the undesired undulations (ears) appearances were located on the rim of the cups during forming.

This study starts from this optimal tool set configuration where process conditions are optimized. The focus is to suppress these two last operations by modifying the initial size and shape of the blank.

Classically, to compensate these ears formation, manufacturers have to realize different trials on the forming process by modifying the initial blank shape. This kind of approach needs time, process mobilization for the trials realization and different cutting blank tools. This paper proposes an alternative solution by piloting the blank geometry by a DCT decomposition (Discrete Cosine Transformation). The stamping part defect is then characterized by a modal decomposition. The different trials are realized on a finite element simulation (Abaqus). The experimental design technique is used to find the influence of the blank modification on the final part shape (ear amplitude and final height). These trials allow to calculate different polynomial relations and it is possible to find the optimal blank shape and size which leads to obtain a good part (good dimension without ears).

Studied part and previous optimal results

The studied part is a cylindrical cup made in only one stamping operation. This particular part has to be welded to another part (not shown here). This welding process forces a small shape defect on the bottom zone (to ensure the quality of the welding operation). The drawing and the associated geometrical tolerancing is indicated in the figure 1. The focus is then to reduce the progression to only one forming operation (OP0 and OP1 of the figure 2). For that, it is necessary to precisely define the initial shape of the blank. There are two different objectives to respect:

• The final part height is equal to 30mm.

• The ear amplitude does not exceed 0.2mm.

Assumption:

In the previous study [START_REF] Ledoux | Numerical optimisation of an industrial multi-pass stamping progression by using the design of experiment method[END_REF], it is shown that the blank modification have no influence on the minimal thickness value. So we assumed that it is possible to find an optimal blank (size and shape) without influence on thickness value.

1.1.

Numerical simulation Abaqus software is used for the numerical simulation in Explicit version [START_REF] Hibbit | ABAQUS/Explicit User's Manual version 6[END_REF]. Shell elements (S4R) with four nodes discretize the blank. There are seven integration points in the thickness. The stamping tool is modeled by rigid surfaces. The main parameters used for simulation are listed in table

Realization of the initial simulation

In the aim of identifying the part defect, the initial simulation of the stamping operation is computed. In this case, the initial blank size is determined by a calculation of the needed area. The possibility to obtain the wished geometry of the part and the minimal flange is directly linked to the initial blank diameter. So we have to estimate its initial diameter D0 by taking into account these following hypotheses:

• H1: the volume of the blank is constant during the stamping operation.

• H2: the thickness is totally constant (the local decrease is compensated by local increase).

A geometrical calculation gives an initial diameter D0 = 116.63mm.

From this nominal blank value and with the optimal tool configuration, we can measure the initial ear amplitude of 1.72mm (see figure 4).

Classical approaches

In the figure 4, we can see the ear deviation on the part border projected along the axis 2. The graph on the right hand side quantifies the ear amplitude. Each segment represents local deviation of each node around the height target (35mm). There are different ways to drive this kind of defects. The classical approach consists in associating a theoretical circle on the circumference point. Then, it is possible to define the extreme points that correspond to the shape defect. The data parameters are the position of the circle (point + vector) and a scalar value (distance between the two extreme points). With this approach, we do not know the form defect and it is impossible to identify the corresponding modification to do in the initial blank shape. Another approach consists in measuring the position of each node and to compare it with the wished position. With these parameters, there are as many distance values as nodes.

Discrete geometry parameterization

We propose particular setting parameters by defining a discrete geometry parameterization of the defect. It is then possible to fit the measured geometry with an open basis of derivated geometries. Thus size, position and form defects can be decomposed in the following basis (several discrete parameterizations are possible). The first one is Fourier type transformation [START_REF] Capello | Harmonic fitting approach for plane geometry measurements[END_REF][START_REF] Summerhays | Optimizing discrete point sample patterns and measurement data analysis on internal cylindrical surfaces with systematic form deviations[END_REF] which is well known by the metrologists for the circularity and straightness defects (2D defects for 1 geometry parameter). In the case of optic lenses (ISO 10110-5), Zernike polynoms define a set of possible defects for a disk geometry (3D defects for 2 parameter geometry). The Discrete Cosinus Transformation [START_REF] Huang | Mode-based Decomposition of Part Form Error by Discrete-Cosine-Transform with Implementation to Assembly and Stamping System with Compliant Parts[END_REF] makes possible to define a set of defects for a rectangular geometry (3D defects for 2 parameter geometry) and is used for image processing.

1.3.3.

Modal description of the defect Those solutions are interesting because they allow the description of all the defects according to the parameterization. We propose a model [START_REF] Formosa | Modal expression of form defects[END_REF][START_REF] Adragna | Analysis of shape deviations of measured geometries with modal basis[END_REF] which can be used on any geometry. This model is based on modal analysis of the target part geometry.

Assumptions:

-Due to anisotropy, the part and the part defects have got two symmetries are symmetrical.

-Form defects are taken into account.

The eigen decomposition modes

As it is expected to characterize the part profile, 2D beam elements are chosen for this eigen decomposition mode. The clearance between the target and the simulated profile is calculated in a deviation vector (V). The modal analysis is used for computing modal basis Q i .

Then, this basis is used for the decomposition of V and for calculating λ i coordinates. These coordinates represent the deviations projected in the modal basis.

General purpose

In linear dynamics, the discretized equations of the movement for a conservative, discrete or continuous system can be written in the general form: 1), M is the generalized mass matrix, K is the generalized stiffness matrix and q a dynamic displacements vector. n is the freedom degrees number of the system. The solutions are the modes q i of a structure which can be decomposed as two functions (space*time):

Mq ¨ +Kq=0 (1) In (
q i =Q i .cos(ω i t) (2)
where Q i is the amplitude vector, and ω i the corresponding pulsations in rad.s -1 . Taking into account their definition, the modes are solutions of the equation:

(K -ω i ² M)Q i =0 (3)
(3) admits n eigen solutions, which are the structure modes. The modal pulsations ω i of the various modes are the roots of the characteristic polynomial:

det(K -ω i ²M)=0 (4)
The vectors of modal deformations Q i are the eigenvectors associated with the pulsations ω i , which form a basis M and K orthogonal in the vector space of the structure movements.

The norm is then ||A||∞ = Max(|Aj|) [START_REF] Box | Empirical Model-Building and response surface[END_REF] The orthogonality of the eigen modes expresses that the inertias and stiffness developed in a mode do not work in the movements of the other modes. There is mechanical independence of the modes. This method gives all the possible defects of the part (mode shapes) through modal shape decomposition. In fact, the discretization limits the number of possible defects.

Application of the method for the studied part

The stamped part geometry results from numerical simulation. It is discretized with 35 elements with equal length. The V clearance vector is calculated from the deviation of the target geometry to the simulated stamped one. The eigen modes of the geometry are identified by modal analysis (see relation 3).

A projection in this base of the V vector is done that allows the calculation of the λ i coordinates. Its decomposition is carried out using the vectorial projection:

λ i = < V.Qi > = 1 ||Qi||² Q i t V (6)
Where λ i is the coordinate associated to the mode Q i , so called the modal coordinates. The figure 5 represents the five first mode shapes and can be compared to the target geometry. In this particular case, it is possible to show that the modal decomposition leads to define a discrete cosine shape with different amplitudes and wavelength for the part defect reconstruction. In this special model, represented in the figure 6, the modal model corresponds to a beam with only one degree of freedom in y axis. It is possible to write then different relations:

Y(x) = A 0 Ch(αx) + B 0 Sh(αx) + C 0 cos(αx) + D 0 sin(αx) (7)
Y (1) Where Y (i) correspond to the i th function derivate.

By using the boundary conditions for this model:

Y (1) (0)= 0, Y (3) (0)= 0 Y (1) (l) = 0, Y (3) (l) = 0

And then, by using the equations 8 and 10:

Y (1) (0) = 0 → B 0 + D 0 = 0 B 0 = D 0 = 0 Y (3) (0) = 0 → B 0 -D 0 = 0 Y (1) (l) = 0 → A 0 Sh(αl) -C 0 sin(αl) = 0 ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ Sh(αl) -sin(αl) Sh(αl) sin(αl) ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ A 0 C 0 = ⎣ ⎢ ⎡ ⎦ ⎥ ⎤ 0 0 Y (3) (l) = 0 → A 0 Sh(αl) + C 0 sin(αl) = 0
And finally,

→ α = nπ l → A 0 = 0
And then, Y(x) can be write as follow:

Y(x) = C 0 cos ⎝ ⎛ ⎠ ⎞ nπ l x (11)
Figure 7 shows the modal decomposition of the nominal geometry. It can be obvious that the first modes are the most contributing to the modal description of the part's defects.

In this paper, the five first modes are used to describe the part's defects (size and shape). The projection residue value is less than one percent. A combination with these different DCT coefficients allows to obtain on the initial blank more complex shape.

Modes i Modal coordinates

Optimization strategy

The optimization of the process parameters by the DOE technique needs to follow these four different steps:

1. Selection of the range variation of each process parameters, 2. Choice of the mathematical model and realization of the experiments, 3. Calculation of the polynomial relations and analysis of their influence on the final part defect 4. Optimization by using the obtained mathematical model and verification

Selection of the range variation of each DCT parameters

According to the previous estimation of the initial blank diameter, the initial simulation and its trends, it is possible to define the range variation of each DCT parameters. These choices are very important and could compromise the well running of the optimization. Two extreme cases can be found. Either the search space is too small then the optimum can be outside, or the search space is too big and the accuracy can be not sufficient.

We suppose that modifying the lobe shape of ±1.5mm, is sufficient to compensate the anisotropy ears (the initial amplitude corresponds to 1.72mm). Moreover, it seems that the initial blank size is very close to the optimal one because the obtained height with the initial simulation is equal to 34.69mm. The variation range of the diameter is then ±1.5mm.

By taking into account these constraints, we choose their variation ranges. Table 3: variation range of each DCT coefficient

Model choice and experimental design

We enhanced the accuracy by selecting a second degree polynomial model which assumes a quadratic variation of the geometrical output from every process input: Y= a 0 +a 1 X 1 + a 2 X 2 +…+ a 12 X 1 X 2 + a ij X i X j +…+ a 11 X 1 2 + a nn X n 2

In this model, the terms X i X j are interactions between factors, which means that the effect of one of them (X i ) depends on the value of the other one (X j ).

A Box-Wilson central composite design (called central composite design or CCD design) was used ( [START_REF] Box | Empirical Model-Building and response surface[END_REF] and [START_REF] Mongomery | Design and analysis of experiments[END_REF]). This is a factorial fractional design (experiments 1 to 8 in table 4), with a central point (experiment 9, table 4). It was enhanced with a group of star points to enable estimation of quadratic curvatures. The distance from the center of the design space to a factorial point was ±1 unit (in normalized variation) for each factor, the distance from the center of the design space to a star point was ±α, with | | α > 1. Τhe precise α value depends on some properties desired for the design and on the number of factors involved. In our case, the optimal value for α was ±1.287, [START_REF] Box | Empirical Model-Building and response surface[END_REF].

It is possible to define the corresponding experimental design for the quadratic polynomial calculation. Each line of the table 4 corresponds to a numerical simulation carried out with the corresponding blank with particular D, A1 and A2 values. The result was a file of points from the bottom contour. This file was post-processed in order to measure the modal coefficients (L1, L2, L3, L4 and L5) of each simulated part. All these results are presented in table 4.

L1 L2 L3 L4 L5 N° exp. D A 1 A 2 P r o f i l e 0 0 0 0 0 <-target values 1 115.5 -1.5 -1.5 Profile 1 -1.35E+00 1.68E+00 8.72E-01 -3.32E-01 -9.07E-02 2 115.5 -1.5 1.5 Profile 2 -1.24E+00 1.65E+00 -1.48E+00 2.56E-01 -6.70E-02 3 115.5 1.5 -1.5 Profile 3 -1.22E+00 -1.03E+00 1.12E+00 -

Calculation of polynomial model coefficients and analysis of their influence

2.3.1.

Model calculation From the results of the experimental design, coefficients of the polynomial model linking the modal coefficients to the DCT coefficients were calculated by the multi-linear regression method. The following quadratic equations were obtained (equations 12, 13, 14, 15 and 16): Where Li' are the standardized variables, respectively corresponding to Li. They range from -1 to +1, while the corresponding variables range from the minimal to the maximal values.

In order to check this mathematical model, a numerical simulation was carried out at the centre of the domain (number 9 in table 4), corresponding to the mean values of the tool parameters. The simulation results were compared to the mean values of quadratic model for each modal coefficient.

2.3.2.

Influence of the DCT coefficients on the modal parameters By using the polynomial relations (equations 12, 13, 14, 15 and 16) we find the most influent parameters (D, A1 and A2) on the modal values (L1 to L5). As it can be seen on the figure 9, the most influent parameter on the L1 defect is D. This parameter directly modifies the initial value of the blank diameter. The A1 is the most influent parameter on the L2 defect. This parameter drives the ovalization (two lobes along the circumference) of the part and that is L2 which models this defect type. And then, A2 is the most significant parameter for the L3 defect. A2 allows to give a four lobes form. The defect L4 and L5 defect is quasi equal to zero.

The modal coefficients (Li) are linked to the blank shape (D or Ai). Moreover, we can underline that all the constant terms are null excepted for the L3 coefficient. That means that, with a circular blank, the main defect corresponds to four lobe defects.

2.4.

Optimal configuration computation The previous polynomial relations (equations 12, 13, 14, 15 and 16) are used to determine the optimal values of the initial blank which minimizes the shape and size defect of the part. For that, a particular optimization function is built. And when the function F (eq 17) is minimized, that means that there is no deviation between the target geometry and the stamped part.

Minimized F = Minimized ⎝ ⎜ ⎜ ⎛ ⎠ ⎟ ⎟ ⎞ ∑ i=1 n eigen mode = 5 (Li)² (17)
In our case, five modes are used for the defect description, so, the F function ( 18) is:

Minimized F = Minimized ( ) (L1) 2 +(L2) 2 +(L3) 2 +(L4) 2 +(L5) 2 (18)
Optimal values of the initial blank are obtained and a numerical simulation by finite element method is carried out.

Quadratic optimum

Results

Table 5: Optimal configuration and modal decomposition of the corresponding part defect.

The corresponding optimum and nominal decompositions are presented in the figure 9. The corresponding size and shape defects are represented in the figure 10. With this optimization, the ear amplitude is divided by ten compared to the ears obtained initially. As we can see, with only fifteen numerical experiments it is possible to improve considerably the blank geometry and then to decrease the number of forming operations (two operations against five initially planed). The geometrical specification (figure 1) is now respected.

Modes i Modal coordinates

Modes i Modal coordinates

If we want to improve this solution, it is necessary to take into account more driving parameters with more than four lobes defects. If we look at the modal decomposition of the optimal part, it can be observed that the most important defect corresponds to the mode L5. As it has been explained by the exploitation of the polynomial relation, there is no DCT coefficient that has an effect on it. A higher level of DCT (A3 and more) would decrease the residue.

Conclusion and outlook

In this paper, it is proposed a new approach for reducing the forming operations in the case of deep drawing. Thanks to a previous result, we have obtained a part with a minimal thickness reduction. This phenomenon corresponds to the main problem in the manufacture of this kind of part. From the initial progression, composed of five steps (blank cutting, forming, flanging and cutting), it is proposed to reduced it to only two (blank cutting and forming). For that, it was developed a compensation of the ear formation, in the aim to obtain at the end of the first forming operation a final product. A modal decomposition will allow us to characterize the ear shape. We defined particular polynomial models linking different blanks (size and shape) generated by discrete cosine transformation to the modal decomposition coefficients. Using these models, an optimization is possible in the aim to find the best blank which leads to minimize the ear apparition (by reducing by ten the ear amplitudes). This approach allows to simplify the industrial forming progression drastically (from five to two) and to minimize the needed material.

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1 : initial target geometry

Figure 4 :

 4 Figure 4 : Initial simulation and corresponding deviations around the target height 1.3. Defect type and corresponding setting parameters 1.3.1.Classical approachesIn the figure4, we can see the ear deviation on the part border projected along the axis 2. The graph on the right hand side quantifies the ear amplitude. Each segment represents local deviation of each node around the height target (35mm). There are different ways to drive this kind of defects. The classical approach consists in associating a theoretical circle on the circumference point. Then, it is possible to define the extreme points that correspond to the shape defect. The data parameters are the position of the circle (point + vector) and a scalar value (distance between the two extreme points). With this approach, we do not know the form defect and it is impossible to identify the corresponding modification to do in the initial blank shape. Another approach consists in measuring the position of each node and to compare it with the wished position. With these parameters, there are as many distance values as nodes.

Figure 5 :

 5 Figure 5 : size and shape modal defects

Figure 6 :

 6 Figure 6: modal model.

  (x) =α (A 0 Sh(αx) + B 0 Ch(αx) -C 0 sin(αx) + D 0 cos(αx)) (8) Y (2) (x) = α 2 (A 0 Ch(αx) + B 0 Sh(αx) -C 0 cos(αx) -D 0 sin(αx)) (9) Y (3) (x) = α 3 (A 0 Sh(αx) + B 0 Ch(αx) + C 0 sin(αx) -D 0 cos(αx))[START_REF] Formosa | Modal expression of form defects[END_REF]

Figure 7 :

 7 Figure 7 : Initial part defect modal coordinates.1.4.Blank shape control by DCT For driving the initial size and shape of the blank, a discrete cosine transformation (DCT) is used. We decide to use three different signals. The first (named D) defines the value of the initial blank diameter, the second (named A1), the ovalization (two lobes) and the third (named A3), the four lobe shape. The figure8shows the different possible shapes.

Figure 8 :

 8 Figure 8 : different blank shapes obtained with D, A1 and A2 parameters.

Figure 9 :

 9 Figure 9: graph of the influence

Figure 9 :

 9 Figure 9: comparison of the optimum and initial defect modal decomposition

Figure 10 :

 10 Figure 10: comparison of the nominal and optimum deviations

Blank discretization Tool discretization Process parametersTable 1 .

 1 1. Numerical parameters of the finite element simulation.

	element type	Shell (4 nodes)
	integration points	7
	number of elements	1428
	tool type	Rigid surface
	Punch speed	3m/s
	friction coefficient	0.1
	The used material is a USB steel sheet (DC05) of 0.8mm thickness. The hardening is determined on
	tensile tests using extensometer and image analysis system for high strain levels [3]. It is introduced
	point by point in the FE code. The used behavior law takes into account the anisotropy according to
	Hill's 48 criterion [4]. The material characteristics are shown in table 2.
	USB (DC05)
	Young modulus	206.62GPa
	Yield Strengh	168.12MPa
	Poisson modulus	0.298
	Density	7800kg/m3
		r0° = 2.09
	Anisotropy coefficients (Lankfort)	r45° = 1.56
		r90° = 2.72

Table 2 .

 2 Main USB material properties.

Table 4 :

 4 Central composite experimental design.

	1.29E-01 -9.22E-02