Chloée Bournaud

Thomas Lecourt

Laurent Micouin
email: laurent.micouin@univ-paris5.fr

Catherine Méliet

Francine Agbossou-Niedercorn

Desymmetrization of meso-Bicyclic Hydrazines by Rhodium-Catalyzed Enantioselective Hydroformylation

Keywords: asymmetric hydroformylation, rhodium, hydrazine, enantioselective catalysis

HAL is

Introduction

The enantioselective transition metal-catalyzed hydroformylation of olefins constitutes one of the most attractive atom economic, clean, and environmentally friendly process for introduction of a formyl residue with a concomitant creation of chirality in an organic compound. [1,2] In this context, lot of efforts have been devoted to the design of phosphorous based chiral ligands in order to modify platinum and rhodium hydroformylation catalysts. [1][2][3][4] From a substrate standpoint, styrene and other vinyl arenes, vinyl acetate, and allyl cyanide, for example, [3,4] have been used in order to evaluate the newly designed catalytic systems. Conversely, norbornene and its related derivatives have been hydroformylated selectively into the exo-formyl products in the presence of platinum and rhodium catalysts modified by chiral diphosphorous auxiliaries. [5,[START_REF] Jensen | During our investigations[END_REF] We have an ongoing interest both in asymmetric transformations involving bicyclic hydrazines to synthesize polyfunctional aminocyclopentanes [7][8][9] and hydroformylation. [10] As such, we have reported recently on the rearrangement, [11,12] hydroboration, [8,13,14] and asymmetric nucleophilic ring opening [15,16] of bicyclohydrazine derivatives. We extended our study on such substrates and explored their behaviour under enantioselective hydroformylation conditions. Here, we report on the desymmetrization though asymmetric hydroformylation of three bicyclic hydrazine substrates.

Results and Discussion

As mentioned above, we have investigated the desymmetrization of polycyclic hydrazines by hydroformylation with the aim of having access to new valuable intermediates for further elaboration into polyfunctional cyclopentanes. Initial experiments have been carried out on substrate S1 (Figure 1). The structural analogy between this compound and norbornene prompted us to first evaluate reactions conditions described by Stille for the norbornene hydroformylation. [5c] However, no hydroformylated product could be obtained in the presence of a platinum catalyst generated from Pt(cod)Cl 2 (1 mol-%), SnCl 2 (2 mol-%), and (S,S)-BPPM (2 mol-%) (Figure 2) under the following conditions: syngas (CO/H 2 1:1) 60 bar, 60 °C, 18 h. The lower initial pressure applied compared to that utilized by Stille for the hydroformylation of norbornene (165 bar) can account for this result. As we planed to apply rather mild reaction conditions (< 100 bar), we favoured the use of rhodium based catalytic systems. [1] The next attempt of hydroformylation of S1 has been performed in the presence of the catalytic system obtained from Rh(acac)(CO) 2 and (S,S)-BPPM (1/4) in toluene at 80 °C under 40 bar of syngas (Scheme 1). The conversion was quantitative as determined by 1 H NMR and the hydroformylated product 2 was isolated in 84% yield after workup as a single exo diastereomer. No hydrogenated derivative could be detected in the crude reaction mixture. As the optical purity determination could not be carried out on the aldehyde 2 because of its partial chemical and configurational instability, the crude was first reduced into the corresponding alcohol 3 (NaBH 4 , MeOH, 0 °C, 80 %) prior to the enantiomeric purity determination by chiral HPLC. An encouraging ee of 44 % was obtained. The absolute configuration of the major enantiomer was assessed by a chemical correlation. Thus, we first performed an enantioselective hydroboration of substrate S1 followed by the trapping of the hydroborated intermediate 4 by trimethylsilyldiazomethane. The homologated boronate was then subjected to an oxydative cleavage, leading to the hydroxy silyl derivative 5 in 95 % overall yield (Scheme 4). [17] Then, a desilylation provided the primary alcohol 3 in 87 % isolated yield and with 80 % ee determined by HPLC. Since the absolute configuration of the hydroboration product 6 resulting from the transformation of S1 under the same catalytic conditions followed by an oxidative cleavage had previously been established to be (1S, 4R, 5R), a (1S, 4S, 5S) configuration could be assigned to alcohol 3. As the retention times for the major isomers obtained following both synthetic routes depicted in schemes 1 and 2 were identical, we deduced that both alcohols possess the same absolute configuration (1S, 4S, 5S). Scheme 2. Chemical correlation for the assignment of the absolute configuration of 3.

The variation of the hydroformylation parameters (ligand/Rh ratio, source of Rh, temperature, pressure, ligand) on the reaction course were then investigated (Table 1). A rhodium/ligand ratio from 1/1.2 to 1/4 delivered close catalytic results, especially in terms of enantioselectivity (entries 1-4). Two rhodium sources (Rh(acac)(CO) 2 and Rh(H)(CO)(PPh 3 ) 3 ) generated catalysts of similar efficiencies whereas [Rh(cod)Cl] 2 proved to be less efficient for this transformation (entries 5-7). The temperature had a significant impact on the selectivity, the highest ee being reached at 20 °C (59 % ee at 20 °C vs. 45 % ee at 80 °C, entries 8-11). Among the four chiral ligands evaluated, (S,S)-BPPM proved to be the most efficient, with a 59 % ee obtained at 20 °C. In our hands, the use of (S,S,R,R)-TangPhos ligand at 60 °C led to the hydroformylated compound with a 53 % enantiomeric excess, whereas a 60 % ee has been reported for this reaction conducted at room temperature by Huang et al during our investigations. [START_REF] Jensen | During our investigations[END_REF] The use of (S,S)-BDPP and (R,S)-BINAPHOS provided selectivities of 13% and 32 % ee, respectively (entries 11-14).

Scheme 3. Hydroformylation of substrates S1 -S3.

From the analysis of the results obtained with substrate S1 while varying the catalytic parameters, we selected (S,S)-BPPM and

S1 -S3

OHC HO (S,S,R,R)-TangPhos ligands to examine the hydroformylation of two other bicyclic hydrazine substrates, S2 and S3 into P2 and P3 after reduction of the intermediate formyl derivatives (Figure 1 and scheme 3). The results are summarized in table 2. Despite their apparent structural similarities, the three substrates proved to behave differently in the hydroformylation reaction. Thus, lowering the reaction temperature led to an improvement of the enantioselectivity when working on substrate S1 (entries 3, 5, 11, and 15) and S3 (entries 23, 24) with BPPM ligand, while it proved to be detrimental for both catalyst rate and selectivity for substrate S2 (entries 18-20). A similar behaviour could be observed with the (S,S,R,R)-TangPhos ligand for substrates S1 and S2 (entries 16, 17 and 21, 22). Substrate S3 was not very soluble in the reaction medium. Thus, the yield into the aldehyde was lower than for the two other substrates S1 and S2. Nevertheless, this substrate led to the best result in terms of selectivity as, after reduction, the corresponding alcohol could be isolated with 77.5 % ee. It appears thus that the desymmetrization of meso hydrazines using catalytic asymmetric hydroformylation reactions is far from being straightforward, and highly substrate-dependent.

Table 1. Asymmetric hydroformylation of S1. [a] Entry Rh source Chiral auxyliary Rh/ligand PH2/CO (bar), Temp. (°C) 3 Yield (%) [b] ee (%) [c] alcool Rh(acac)(CO)2 (S,S,R,R)-TangPhos 1/2 30 60 84 53 14 [d] Rh(acac)(CO)2 (R,S)-BINAPHOS 1/4 30 50 86 32

[a] General conditions of hydroformylation: Rh precursor 1 mol-%, toluene 5 mL, non optimized reaction times of 15 or 18 h. [b] The conversion of S1 is total in all cases. Yield of the isolated product 3 obtained by reduction of 2. [c] ee determination was carried out on the alcohol by chiral HPLC, configuration of the major enantiomer (1S,4S,5S), see text. [d] Reaction time of 4 h.

Table 2. Asymmetric hydroformylation of S1 -S3. [a] Entry Substrate L* PH2/CO (bar) T (°C) Time (h) Conv. (%) [b] Yield (%) [c] ee (%) [d] .5 [e] [a] General conditions of hydroformylation: Rh(acac)(CO)2 1 mol-%, chiral auxiliary 2 mol-%, toluene 5 mL. [b] Conversion of S1 -S3 determined by 1 H NMR. [c] Yield of the isolated reduction products. [d] ee determination was carried out on the alcohol by chiral HPLC, configuration of the major enantiomer (1S,4S,5S), see text [e] Absolute configuration not determined.

Conclusion

In summary, we have shown that a desymmetrization of mesobicyclic hydrazines is possible through asymmetric hydroformylation. The reaction proceeds efficiently with a high diasteroselectivity as the exo-products are obtained exclusively. The alcohols prepared by the reduction of the hydroformylated products could be obtained with an ee up to 77.5 % under the catalytic conditions examined in this study. This reaction seems to be much more substrate-dependent than the hydroboration, and its optimisation is still a challenging task.

Experimental Section

1 H NMR, 13 C NMR spectra were recorded on a BRUKER Avance 400 or 300 spectrometer at 25°C. Chemical shifts are reported in ppm downfield of internal tetramethylsilane. All melting points are uncorrected. All reagents are commercially available and were used without further purification. The organic syntheses were carried out under classical conditions unless otherwise stated. The reactions are performed under nitrogen. THF was dried over CaCl2, filtrated through basic alumina and distilled over Na under nitrogen. Toluene was distilled over Na under nitrogen. Substrates S1-S3 were prepared following reported procedures. [15] General procedure for hydroformylation. Under nitrogen, a Schlenk flask was charged with a magnetic stir bar, Rh(acac)(CO)2 (2.6 mg, 0.01 mmol), the chiral auxiliary (0.02 mmol), and toluene (5 mL). The solution was stirred at room temperature for one hour. The substrate (1 mmol) was placed in a double walled stainless steel autoclave equipped with a stir bar under nitrogen. The solution containing the rhodium precatalyst was transferred into the autoclave via cannula under nitrogen. The autoclave was then pressurized with syngas (CO/H2 1:1) at the selected pressure and placed at the desired temperature. At the end of the reaction, the autoclave is cooled to room temperature and depressurized. Toluene is evaporated under reduced pressure. The crude residue is dissolved in methanol and the mixture is cooled to 0 °C. NaBH4 (113 mg, mmol) was added in small portions over 30 min. Then, ethyl acetate (30 mL) and saturated ammonium chloride (10 mL) were added. The aqueous phase was extracted with ethyl acetate (2 x 30 mL). The combined organic layers were washed with brine (20 mL) and dried over magnesium sulfate. After filtration and evaporation of the solvent, the crude residue was purified through silica gel flash chromatography. 3, 36.2, 52.3, 60.5, 68.3, 128.1, 128.4, 128.7, 135.9, 157.2, 199.1.

Dibenzyl

5-(Hydroxymethyl)-2,3-diaza-bicyclo[2.2.1]heptane-2,3dicarboxylate (3, P1)

Yellow oil (364 mg, 80 %). 1 H NMR (400 MHz, CDCl3):  [ppm]1.00-1.30 (m, 1H), 1.50-1.70 (m, 2H), 1.80-2.00 (m, 1H), 2. 10-2.40 (m, 2H), 3.20-3.40 (m, 1H), 3.40-3.60 (br. s, 1H), 4.40-4.80 (m, 2H), 5.10-5.30 (m, 4H), 7.26-7.32 (m, 10H). 13 C NMR (100 MHz, CDCl3):  [ppm] 32. 5, 35.3, 42.7, 60.5, 62.1, 64.2, 68.1, 128.1, 128.3, 128.6, 136.1, 157.7. HRMS calcd for C22H25N205: 397.1763, found 397.1765. Conditions for chiral HPLC analysis: Samples were injected in elution solvent at a 0.1gL -1 concentration in a 20 mL injection loop. Analysis conditions: Chiralpack AD column (0.46 cm I.D. x 25 cm) equipped with a precolumn (0.46 cm I. D. x 5 cm), detection at  = 220 nm, Flow = 0.8 mL.Min -1 , eluent : hexane/iPrOH 70/30, retention time: tr1 = 11.87 min, tr2 = 13.81 min.

3,5-Dioxo-4-phenyl-2,4,6-triazatricyclo[5.2.1.0 2,6 ]decane-8carbaldehyde White amorphous solid (270 mg, 99 %). 1 H NMR (300 MHz, CDCl3): 

[ppm] 1.70 (d, J = 11.0 Hz, 1H), 1.91 (d, J = 11.0 Hz, 1H), 2.15 (dd, J = 11.5, 9.0 Hz, 1H), 2.29 (m, 1H), 3.26 (m, 1H), 4.99 (s, 1H), 5.31 (s, 1H), 7.40 (m, 1H), 7.47 (m, 4H), 9.78 (s, 1H). 13 C NMR (100 MHz, CDCl3): 

[ppm] 28. 7, 36.9, 52.4, 60.1, 125.5, 128.6, 129.4, 131.4, 157.0, 198. 5, 36.2, 43.0, 60.3, 61.9, 63.7, 125.6, 128.5, 129.3, 131.6, 157.0. MS (ES) 296 (MNa + ) Conditions for chiral HPLC analysis: Samples were injected in elution solvent at a 0.1gL -1 concentration in a 20 mL injection loop. Analysis conditions: Chiralpack AD column (0.46 cm I.D. x 25 cm) equipped with a precolumn (0.46 cm I. D. x 5 cm), detection at  = 220 nm, Flow = 0.8 mL.Min -1 , eluent : hexane/iPrOH 70/30, retention time: tr1 = 11.9 min, tr2 = 13.8 min. 8, 35.4, 43.7, 58.6, 60.4, 63.9, 127.3, 129.9, 133.1, 153.4, 153.7. MS (ES) 281 (MNa + ) Conditions for chiral HPLC analysis: Samples were injected in elution solvent at a 0.1gL -1 concentration in a 20 mL injection loop. Analysis conditions: Chiralpack AD column (0.46 cm I.D. x 25 cm) equipped with a precolumn (0.46 cm I. D. x 5 cm), detection at  = 220 nm, Flow = 0.8 mL.Min -1 , eluent : hexane/iPrOH 80/20, retention time: tr1 = 14.18 min, tr2 = 15.56 min.

2-(Hydroxymethyl

Chemical correlation

[Rh(cod)Cl]2 (5 mg, 0.01 mmol), (S,S)-BDPP (8.8 mg, 0.02 mmol) and S1 (365 mg, 1 mmol) were placed in a Schlenk tube, dried under vacuum (0.1 mmHg) for 1 h, and then placed under argon. DME (4 mL) was degassed at -50 °C and added to the mixture at this temperature. The yellow-green slurry was stirred at -50 °C for 30 min. Catecholborane (210 µL, 2 mmol) was then added dropwise, and the mixture became orange but remained heterogeneous. The reaction was kept at -50 °C for an additional 30 min. Solvent and excess reagent were then carefully removed under vacuum (0.1 mm Hg, 3 h) to give the intermediate borane 4 as a dark yellow foam. A solution of 4 in THF (6 mL) under argon was then added over a 2 M solution of trimethylsilyldiazomethane in Et2O (2.5 mL, 5 mmol). After refluxing overnight, 10 mL of a freshly prepared 1:1 mixture of 2N aqueous sodium hydroxide and 30 % hydrogen peroxide were then added dropwise at 0 °C, turning the solution to black, and the mixture was stirred for an additional 4 h at room temperature. After extraction with EtOAc (3×20 mL), the combined organic layers were washed with 1M HCl (20 mL), dried over MgSO4, filtered and concentrated. The crude reaction mixture was then purified by silica gel flash chromatography (cyclohexane / ethylacetate 2:1) to give 5 as a mixture od diastereomers (446 mg, 95 %). Tetrabutylammonium fluoride (250 mg, 0.8 mmol) was added to a solution of 5 (188 mg, 0.4 mmol) in THF (2 mL) and stirred at room temperature for 24 h. More reagent was then added (250 mg, 0.8 mmol), and the mixture was kept at room temperature until complete consumption of the starting material. After concentration under vacuum, the crude reaction mixture was purified by silica gel flash chromatography to give 3 (138 mg, 87 %). The absolute configuration of compound 3 was established from the known configuration of compound 6, obtained by an oxidative treatment of boronate 5 (see ref 7).

Figure 1 .

 1 Figure 1. Substrates for hydroformylation studies.

Figure 2 .

 2 Figure 2. Chiral ligands used in this study.

  Yellow oil (330 mg, 84 %). 1 H NMR (400 MHz, CDCl3):  [ppm] 1.46 (d, J = 10.5 Hz, 1H), 1.69 (d, J = 10.5 Hz, 1H), 1.81-2.20 (m, 2H), 2.99 (br. s, 1H), 4.51-5.15 (m, 2H), 5.10-5.27 (m, 4H), 7.26-7.34 (m, 10H), 9.46 (br. s, 1H). 13 C NMR (100 MHz, CDCl3):  [ppm] 29.

  White solid (245 mg, 90 %). M.p. = 180-182 °C. 1 H NMR (300 MHz, CDCl3):  [ppm] 1.47 (ddd, J = 13.0, 5.0, 3.0 Hz, 1H), 1.94 (m, 2H), 2.11 (m, 2H), 2.41 (m, 1H), 3.44 (m, 1H), 3.64 (m, 1H), 4.68 (br.s, 1H), 4.88 (br.s, 1H), 7.40 (m, 1H), 7.47 (m, 2H), 7.49 (m, 2H). 13 C NMR (100 MHz, CDCl3):  [ppm] 31.

  Yellow oil (125 mg, 42 %). 1 H NMR (400 MHz, CDCl3):  [ppm] 1.66 (m, 1H), 2.08 (sl, 2H), 2.15 (m, 1H), 2.45 (m, 1H), 3.53 (m, 1H), 3.69 (br.s, 1H), 3.75 (m, 1H), 5.37 (br.s, 1H), 5.43 (br.s, 1H),7.76 (m, 2H), 8.26 (m, 2H).

  13 C NMR (100 MHz, CDCl3):  [ppm] 31.

Acknowledgments

We thank Pr Kyoko Nozaki for a generous gift of BINAPHOS. We thank the "Centre National de la Recherche Scientifique" and the "Ministère de la Recherche et de l'Enseignement Supérieur" (grant to Chloée Bournaud) for financial support. ____________

Entry for the Table of Contents (Please choose one layout)

Layout 2:

((Asymmetric Hydroformylation)) Chloée Bournaud, [a] Thomas Lecourt, [a] Laurent Micouin, [a],* Catherine Méliet, [b] Francine Agbossou-Niedercorn [b],* …….... Page No. -Page No.

Desymmetrization of meso-bicyclic hydrazines by Rhodium catalyzed enantioselective hydroformylation

Keywords: asymmetric hydroformylation, rhodium, hydrazine

Asymmetric hydroformylation of three meso-bicyclic hydrazines followed by the reduction of the formyl product yielded the corresponding desymmetrized optically enriched hydroxymethyl hydrazines (up to 77.5 % ee).


Alejandro Pérez Luna

Martine Bonin

Laurent Micouin

Henri-Philippe Husson

Reversal of Enantioselectivity in the Asymmetric Rhodium-versus Iridium-Catalyzed Hydroboration of meso Substrates

Since Männig and Nöth first reported the transition metal catalyzed hydroboration, 1 increasing attention has been paid to the development of this new reaction, especially in the field of asymmetric catalysis. [START_REF] Hayashi | Comprehensive Asymmetric Catalysis[END_REF] It was found that chemo-, 1 regio-3 and diastereoselectivities 4 could be obtained opposite to the uncatalyzed processes. Furthermore, beside the access to enantiopure alcohols or amine derivatives, the transmetalation of the intermediate boronic esters to the corresponding organozinc derivatives considerably broadens the scope of the rhodiumcatalyzed hydroboration. 5 However, applications of this promising reaction have been restricted to a limited number of substrates; only successful enantioselective hydroboration of styrene and its derivatives (ee > 95 %), and norbornene in a lesser extent (ee = 84 %) have been reported. [START_REF] Hayashi | Comprehensive Asymmetric Catalysis[END_REF] Despite intensive work, no other metallic substitute for rhodium has been reported to be effective in such an asymmetric transformation. Moreover, improvement of this process is hampered by the understanding of its mechanism which is rather vague. 6 Herein we report the first example of asymmetric iridium-catalyzed hydroboration. We also bring new insight into the catalytic cycle of this reaction.

Recently, we have developed a straightforward access to polysubstituted diaminocyclopentanes 3 based on the desymmetrization of meso bicyclic hydrazines 1 via rhodiumcatalyzed asymmetric hydroboration (Figure 1). Although very few examples of racemic iridium-catalyzed hydroborations have been reported and the reaction with endocyclic alkenes is believed to be very sluggish, 8 the enhanced reactivity of our substrates compared to norbornene prompted us to test iridium catalysts with this series (Table ). Using our previously established hydroboration conditions, we were pleased to see that alcohol 2a could be obtained with 1 mol% of [Ir(COD)Cl] 2 as iridium source (Entry 2). A higher chemical yield could be achieved using the same amount of precatalyst loading by increasing the reaction duration (Entry 3). Reaction at room temperature with DIOP as a ligand could afford compound 2a in 40% yield with 44% ee (Entry 5). This results show for the first time that, despite a lower reactivity, neutral iridium complexes can catalyze asymmetric hydroboration. More interestingly, a complete reversal of enantioselectivity was observed between Rh and Ir systems. This phenomenon appeared as a general feature since it was observed on all substrates and conditions studied (Scheme 1). To our knowledge, reversal of enantioselectivity in the course of an asymmetric reaction catalyzed by two transition metals having the same d shell electronic configuration is unprecedented. [START_REF]Reversal of enantioselectivity in the asymmetric hydrogenation using Rh-BINAP or Ru-BINAP systems is well documented, but the two catalytic cycles involve completely different reactive intermediates and kinetic profiles[END_REF] This observation brings new information on the mechanism of hydroboration. Two main catalytic cycles, based on experimental and theoretical studies, have been proposed (Figure 2).

6b While several authors describe a key step involving a migratory insertion into the metal-H bond, [START_REF] Evans | [END_REF] others favor the migration of a boryl group.

11 If the insertion step is the determining one in the cycle, these two pathways should lead to opposite regioisomers or, in the case of meso substrates, enantiomers. The possible substrate-dependent reversibility of this step complicates the analysis. [START_REF] Evans | [END_REF] In our case, we have a dramatic acceleration of the hydroboration with bicyclic hydrazines compared to the norbornene case. This can be explained by the stabilization of the transition state of the migratory insertion by two σ*-nitrogen lone pair interactions, and it suggests that this is the first irreversible step in the cycle (Figure 3). Thus, solvent-and ligand-dependent stereoselectivity can be explained by means of the difference in diastereomeric transition states energetic levels (TS 1 versus TS 1' and TS 2 versus TS 2' ), and systematic reversal of enantioselectivity (regioselectivity) is likely to be the result of two different insertion pathways (TS 1 versus TS 2 and TS 1' versus TS 2' ). Theoretical studies on bond dissociation energies have shown that insertion into the Ir-Bcat bond is favored by 8 kcal.mol -1 over the insertion into the Ir-H bond. 13 In the case of our substrates, we can propose that this pathway is involved with [Ir(COD)Cl] 2 and that the use of [Rh(COD)Cl] 2 involves a key Rh-H migratory insertion step.

M P P H Cl BCat M P P H Cl BCat R 1 R 2 M P P Cl BCat R 1 R 2 M P P Cl H R 1 R 2 BCat M P P Cl M P P Cl CatBH CatBH R 1 R 1 R 2 BCat M-B insertion M-H insertion BCat R 2
Finally, chemical yield and enantioselectivity of Ir-based desymmetrization were increased by using commercially available (R,S)-Josiphos ligand, leading to alcohols 2a,b in 46-65 % yield with ee up to 64 % (Scheme 2). These promising results might be improved by a fine electronic tuning of ferrocenyl phosphines or P,N ligands. In summary, we have reported the first Ir-catalyzed asymmetric hydroboration with ee and chemical yields up to 64%. We have also shown an unprecedented reversal of enantioselectivity for a catalytic desymmetrization reaction by exchanging two transition metals having the same d shell electronic configuration. These results helped us to show that, at least on our substrates, Rhcatalyzed hydroboration involves a metal-H insertion and a boryl migration is involved when using an Ir precatalyst. More generally, the use of meso substrates, leading to enantiomers instead of regioisomers, avoids the introduction of thermodynamic bias for regioselectivity, and could be a general tool for the investigation of this kind of problems. 15

7

 7

Figure 1 .

 1 Figure 1. Desymmetrization of meso hydrazines.

a

  Reagents and conditions: (a) M= Rh, 1% [Rh(COD)Cl] 2 , 2% (S,S)-BDPP, CatBH (2 equiv.), DME, -50 °C, 0.5 h,then H 2 O 2 , 50%. M = Ir, 1% [Ir(COD)Cl] 2 ,(S,S)-BDPP, CatBH (2 equiv.), THF, RT, 7 h, then H 2 O 2 , 26%; (b) M= Rh, 1% [Rh(COD)Cl] 2 , 2% (S,S)-BDPP, CatBH (2 equiv.), DME, -50 °C, 0.5 h, then H 2 O 2 , 78%. M = Ir, 1% [Ir(COD)Cl] 2 ,(S,S)-BDPP, CatBH (2 equiv.), THF, RT, 7 h, then H 2 O 2 , 9%; (c) M= Rh, 1% [Rh(COD)Cl] 2 , 2% (S,S)-BDPP, CatBH (2 equiv.), DME, -50 °C to RT, 7.6 h, then H 2 O 2 , 68%. M = Ir, 10% [Ir(COD)Cl] 2 ,(S,S)-BDPP, CatBH (2 equiv.), THF, RT, 7 h, then H 2 O 2 , 17%.

Figure 2 .

 2 Figure 2. Simplified postulated catalytic cycles for hydroboration

Figure 3 .

 3 Figure 3. Possible transition states for the migratory insertion steps. Some groups have been omitted for clarity.

Table .

 . Rhodium-versus iridium catalyzed hydroboration of compound 1a. Influence of ligand.

	N N O	1a O OBn OBn	1) 1% [M(COD)Cl] 2 2% L* 2 equiv. CatBH 2) H 2 O 2	HO		N N CO 2 Bn CO 2 Bn 2a
	entry	L*	solvent	M	yield	er	b
			T(°C) (t[h])		(%)	a
	1	(S,S)-BDPP	DME,-50 (0.5)	Rh	91		92:8	c
	2	(S,S)-BDPP	DME,0-RT (0.5)	Ir	7		32:68
	3	(S,S)-BDPP	DME,0-RT (7)	Ir	30		34:66
	4	(S,S)-DIOP	DME,-50 (0.5)	Rh	46		23:77
	5	(S,S)-DIOP	DME,RT (7)	Ir	40		72:28
	6	(R)-Quinap	DME,-50 (0.5)	Rh	11		62:38
	7	(R)-Quinap	THF,RT (7)	Ir	27		44:56

a Yield of analytically pure compound. b Determined by chiral HPLC. c The absolute configuration of the major enantiomer has been established to be (1R,4R,5R) (See ref 7). Scheme 1. Rhodium-versus iridium catalyzed hydroboration (Absolute configurations not determined).

Acknowledgement. We thank CNRS for financial support.

Supporting Information Available : experimental procedures and analytical data for ee determination. This material is free of charge via the Internet at http://pubs.acs.org.

HO

Yields up to 91% ee up to 84%

Yields up to 65% ee up to 64%

The first example of Ir-catalyzed asymmetric hydroboration with ee and chemical yields up to 64% is reported. The switch from rhodium to iridium leads systematically to opposite enantiomers in this desymmetrization reaction.


Elsa Piollet

Guilhem Michon

Dominique Poquillon

Nonlinear Vibration Behavior of Sandwich Beams with Entangled Fiber Core Material

In this paper, the use of entangled cross-linked fibers as core material in vibrating sandwich beams is investigated. The aim is to analyze the effect of this specific core material in terms of damping. The dynamic shear properties of the material are first studied experimentally. The shear modulus is shown to decrease with increasing shear strain amplitude at low shear strains. To include an amplitude dependency of the core material properties in the sandwich beam behavior, an analytical model is proposed. The equations of motion are derived using Lagrange's equations. The shearing of the core is introduced in the equations through the use of virtual work to allow any relationship between shear stress and shear strain, including damping and nonlinearities. Experimental tests are carried out on sandwich beams with entangled fiber core material. The Frequency Response Function obtained exhibits decreasing resonant frequency and peak amplitude with increasing load amplitude. This softening behavior is consistent with the decreasing shear modulus. The proposed model is used take into account the softening nonlinearity. The FRF is reproduced with a linearly decreasing shear modulus and linearly increasing loss factor.

Introduction

Sandwich panels are widely used in aerospace and automotive structures. They are made of a core bonded to two stiff facesheets. By separating the two facesheets with the core, the bending stiffness is increased as compared to a single-material structure of the same weight. While this high stiffness-to-weight ratio is a strong advantage from a static point of view, it also makes the structures prone to transmit vibrations. In aerospace and automotive fields, unwanted vibrations can lead to user discomfort or damage to the structures, and vibration damping is thus a strong concern. The theory of vibration of damped three-layer beams and panels was considered by a number of authors in the past 50 years. Kerwin [START_REF] Kerwin | Damping of flexural waves by a constrained viscoelastic layer[END_REF] analysed the use of constrained layer to damp flexural waves. DiTaranto [START_REF] Ditaranto | Theory of vibratory bending for elastic and viscoelastic layered finite-length beams[END_REF] and Mead and Markus [START_REF] Mead | The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions[END_REF] proposed a model for sandwich beams with arbitrary conditions and a viscoelastic core, and set the hypotheses that are still in use in the literature. The use of the energy principle (Hamilton's principle or Lagrange's equations) to derive the equations of motion was introduced by Rao [START_REF] Rao | Frequency and loss factors of sandwich beams under various boundary conditions[END_REF].

The most commonly used core material is honeycomb, which does not provide much damping. Control must then be added through either active or passive damping. Active damping consists in the use of sensors and actuators, which need energy input. Passive damping consists in using adapted materials to increase the overall damping of the structure. A layer of viscoelastic material can be added to the structure, either by covering the structure with a constrained or unconstrained viscoelastic layer, or by including the viscoelatic layer between the facesheet and the core [START_REF] Fotsing | Lightweight damping of composite sandwich beams: Experimental analysis[END_REF]. Another possibility lies in considering new core materials to replace totally or partially the classical honeycomb material, as by combining honeycomb and viscoelastic material for example [START_REF] Jung | A combined honeycomb and solid viscoelastic material for structural damping applications[END_REF]. This paper studies the flexural vibration of sandwich beams with a recently introduced core material. Mezeix et al. [START_REF] Mezeix | Mechanical behavior of entangled fibers and entangled cross-linked fibers during compression[END_REF] developed a new core material made of short fibers of glass, carbon or aramid, entangled and cross-linked with epoxy resin. Some vibrations experiments have been carried out by Mezeix [START_REF] Mezeix | Développement de matériaux d'âme pour structures sandwich à base de fibres enchevêtrées[END_REF] and Shahdin [START_REF] Shahdin | Fabrication and mechanical testing of glass fiber entangled sandwich beams: A comparison with honeycomb and foam sandwich beams[END_REF], but this is the first study focused on the vibration properties and modeling of sandwich beams with this core material.

Previous models of sandwich beams considered frequency dependent materials, or geometrical nonlinearities, but no inclusion of amplitude dependent material behavior could be found in the literature.

In this paper, the amplitude-dependency of the dynamic shear properties of the entangled cross-linked fiber material is observed experimentally. An analytical model of the flexural vibration of sandwich beams is derived using Lagrange's equations to include core material nonlinearity. A sandwich beam with entangled cross-linked fibers as core material is studied experimentally. The proposed model is used to reproduce the observed frequency response, and analytical and experimental results are compared.

Entangled cross-linked fiber material description

After studies of Poquillon et al. [START_REF] Poquillon | Experimental data about mechanical behaviour during compression tests for various matted fibres[END_REF] on entangled fibers, Mezeix et al. [START_REF] Mezeix | Mechanical behavior of entangled fibers and entangled cross-linked fibers during compression[END_REF] introduced a material in which short entangled fibers are cross-linked with epoxy resin. Contrarily to most architectured materials, entangled crosslinked fibers have no periodicity: the fiber orientations are randomly distributed.

Different types of fibers can be used and previous studies considered carbon, aramid and glass fibers [START_REF] Shahdin | Fabrication and mechanical testing of glass fiber entangled sandwich beams: A comparison with honeycomb and foam sandwich beams[END_REF][START_REF] Mezeix | Mechanical behavior of entangled fibers and entangled cross-linked fibers during compression[END_REF]. In this paper, the material is made with carbon fibers which are commonly used in aeronautical applications due to their high performances in terms of Young's modulus and tensile strength. The fibers consist in a yarn of stranded carbon filaments. The diameter of the filaments is 7 µm with a Young's modulus of 240 GPa.

The fibers are first cut to a length of 40 mm. To achieve a low density material, the filaments of the yarn must be separated. A blower room is used to obtain the separation and to entangle the fibers, with an air flux of 4 bar pressure.

Then, the entangled fibers are cross-linked with epoxy resin. Epoxy resin is chosen because of the common use of carbon/epoxy prepreg as facesheets in aeronautical sandwich structures. A paint spray gun is used to spray the resin.

Finally, the fiber network is compressed to fit in the mold and polymerized. Figure 1 shows a sample of the material after polymerization.

The volume fraction of the material is defined as the ratio of the entanglement density (before cross-linking) to the density of the bulk material:

f = ρ entanglement ρ bulk (1)
In the present paper, the density of the entanglement is 150 kg/m 3 Material identification

Set-up and samples

This paper focuses on the use of entangled cross-linked fibers as sandwich core material in flexural vibration. As in a large part of the literature [START_REF] Ditaranto | Theory of vibratory bending for elastic and viscoelastic layered finite-length beams[END_REF][START_REF] Mead | The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions[END_REF][START_REF] Cai | Vibration analysis of a beam with PCLD patch[END_REF], the core material is supposed to be deformed only in shear. The material is thus characterized in shear only.

The samples have a thickness of 20 mm, a length of 60 mm and a width of 40 mm. One sample is shown on Fig. 1. The samples are bonded to 3 mm thick aluminum plates on both sides. To ensure that the applied deformation is in shear only, two samples are tested together. A vertical excitation is applied between the two samples by a 100N shaker. The vertical displacement of the central aluminum plates is measured with an inductive displacement sensor and the force is measured with a piezoelectric load cell located at the interface between the shaker and the aluminum plates. The set-up for dynamic shear testing is presented on Fig. 2.

The shear strain γ and shear stress τ are related to the measured displacement u and load F as follows:

γ = u h (2)
τ = F S 2 (3)
where h = 20 mm is the thickness of the sample and S 2 = 2 × 40 mm × 60 mm the area through which the load is applied, i.e. the bonded area between the two samples and the central aluminum plates.

Viscoelastic model

The most classical way to model material including stiffness and damping is to use a viscoelastic modeling, as described by Nashif et al. [START_REF] Nashif | Vibration Damping[END_REF] and Lakes [START_REF] Lakes | Viscoelastic measurement techniques[END_REF] among others.

In case of harmonic loading, the expression of the oscillatory shear strain is:

γ = γ 0 sin(ωt) (4)
Then in the case of linear viscoelastic behavior, the resulting shear stress is also sinusoidal with a phase lag φ: which is related to the shear strain amplitude by the expression:

τ = τ 0 sin(ωt + φ) (5
τ = G γ 0 sin(ωt) + G γ 0 cos(ωt) (6)
or

τ = G γ + G ω γ (7)
where G = G tan(φ) and ˙= d dt . G is the storage modulus, G is the loss modulus, and η = tan(φ) is the loss factor.

These expressions are often used in a complex form with γ = γ 0 exp(iωt). Then τ = G * γ with

G * = G + iG = G (1 + iη), i = √ -1.
Experimentally, G and η are measured from stress-strain loops by using the following relationships:

G = τ max γ max (8) η = τ (γ = 0) τ max (9)
In small deformations, viscoelastic models are generally used with G and η being frequency-dependent but amplitude-independent. Stress-strain loops are then elliptic. However, these models are also used as approximations to describe an amplitude-dependent behavior by defining the storage modulus and loss factor as functions of the amplitude, see [START_REF] Pritz | Non-linearity of frame dynamic characteristics of mineral and glass wool materials[END_REF] for example. When the nonlinearity increases, the stress-strain loop is no longer elliptic and other models must be used.

Parameter identification

Measurements are made at 20 Hz to stay in a quasi-static configuration. In this paper, the focus is on the amplitude dependency. The dissipation is expected to come mainly from friction between fibers, which is a frequencyindependent phenomenon. The hypothesis of amplitude-dependent and frequency-independent material is consistent with the work of Pritz on other fibrous materials such as mineral wool [START_REF] Pritz | Frequency dependance of frame dynamic characteristics of mineral and glass wool materials[END_REF][START_REF] Pritz | Non-linearity of frame dynamic characteristics of mineral and glass wool materials[END_REF].

The range of study is from 5.10 -5 to 1.10 -3 for γ, corresponding to 1.10 -3 mm to 2.10 -2 mm for the vertical displacement u.

Figure 3 shows the evolution of the shear modulus with the shear strain amplitude. The modulus descreases with the amplitude. To describe this evolution, a linear relationship is used: For γ = 5.10 -4 and γ = 1.10 -3 , a loss factor of η = 0.06 is measured. At lower amplitudes, the loss factor is too small to be measured.

G = 1.0 • 10 7 -1.3 • 10 9 | γ | (10)

Sandwich beam model 4.1 Geometry and hypotheses

The system studied is a clamped-free sandwich beam with an electrodynamic shaker modeled by its moving mass and stiffness as shown on Fig. 4.

The sandwich beam is of length L and width B. It is composed of three layers of thickness h β and density ρ β with β = t, b or c, respectively, for the top layer, bottom layer and core layer. The beam is clamped at x = 0, and free at x = L.

The theoretical developments presented here are based on assumptions made by Mead and Markus [START_REF] Mead | The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions[END_REF] and commonly accepted in the literature:

• Displacements are small and occur in the xz plane only.

• Laminates are considered purely elastic, with Young moduli E t and E b . Laminates are treated as Euler-Bernoulli beams: shear strains and rotary inertia are neglected.

• Transverse direct strains in both core and face-plates are also neglected: for a given position x, transverse displacement w in the z direction is identical in the three layers, which are assumed to be perfectly bonded.

Bonding layer thickness is considered to be null.

• Shear stress and shear strain are constant across the depth of the core.

There is no assumption made on the core material behavior at this point.

The geometry of the beam and notations taken are illustrated on Fig. 5. The longitudinal x-wise displacements of the mid-planes of the face-plates are u t (x, t) and u b (x, t). The transverse z-wise displacement of the system is w(x, t).

The shear strain γ in the core is related to the displacements as follows:

γ = u t -u b h c + d h c ∂w ∂x (11)
where

d = h c + 1 2 (h t + h b ) (12)
and the longitudinal displacement at the middle of the core is given by:

u c = 1 2 u t + u b + h t -h b 2 ∂w ∂x (13)

Energies and work

4.2.1 Kinetic energy of the system.

The kinetic energy of the system results from the transverse and longitudinal motion of the three layers, neglecting rotational kinetic energy, and the moving mass of the shaker (subscript s):

T = T t + T b + T c + T s (14)
where

T β = 1 2 ρ β h β B L 0 ∂u β ∂t 2 + ∂w ∂t 2 dx (15)
for β = t, b and c, and

T s = 1 2 m s ∂w ∂t (x f ) 2 (16)
4.2.2 Potential energy of the system.

The potential energy of the system is due to the deformation of the two elastic laminates and the deformation of the shaker's spring:

V = V t + V b + V s (17)
where

V α = 1 2 Lα E α A α ∂u α ∂x 2 + E α I α ∂ 2 w ∂x 2 2 dx (18)
for α = t and b, where E α is the Young's modulus of the laminate, A α = h α B the cross-sectional area and I α the second moment of area, and

V s = 1 2 k s w 2 (19)

Virtual work in the core.

To include any shear behavior of the core, and in particular nonconservative and nonlinear behavior, the contribution of the core to the energy of the system is written as the virtual work of the shear stress:

δW c = 1 2 V τ δγdV (20)
By using the hypothesis of a constant shear stress through the thickness and width of the beam, the expression is simplified to: The virtual work of an external transverse load f (x, t) is:

δW c = 1 2 Bh c L 0 τ δγdx (
δW f = L 0 f (x, t)δw(x, t)dx (22)
4.2.5 Assumed modes.

In order to write Lagrange's equations of motion, generalized coordinates need to be defined. The assumed mode method is used to discretize the continuous system. Three sets of admissible functions are used to discretize the transverse displacement of the beam and the two longitudinal displacements of the laminates. The use of a third set of admissible functions for u t was introduced by Cai et al. [START_REF] Cai | Vibration analysis of a beam with PCLD patch[END_REF]. The time and space dependency are separated and a vector notation is used to obtain the following relationships:

w(x, t) = nw i W i (x)ψ i (t) = W T ψ (23) u b (x, t) = n b j U b,j (x)ξ b,j (t) = U T b ξ b (24) u t (x, t) = nt k U t,k (x)ξ t,k (t) = U T t ξ t (25)
The displacement functions W i , U b,j and U t,k are sets of admissible functions, ψ i , ξ b,j and ξ t,k are the new generalized coordinates, and n w , n b and n t are the number of modes selected respectively for the transverse motion of the system, the longitudinal motion of the bottom layer and the longitudinal motion of the top layer. If n → ∞ for each of the different displacements, the response is considered to be exact. Superscript T denotes the transpose of a vector. Classical mode shape functions [START_REF] Blevins | Formulas for Natural Frequency and Mode Shape[END_REF] are chosen as admissible functions, as they satisfy all geometric boundary conditions. In this study, the three layers are clamped-free beams. For transverse displacement of a clamped-free beam the corresponding mode shape function is:

W i (x) = cosh(λ i x) -cos(λ i x) -σ i [sinh(λ i x) -sin(λ i x)] (26)
where

σ i = sinh(λ i L) -sin(λ i L) cosh(λ i L) + cos(λ i L) (27)
and λ 1 L = 1.875, λ 2 L = 4.694, λ i L = (i -0.5)π for i = 3, ..., n w where L is the length of the beam.

For longitudinal displacement in a clamped-free configuration:

U t,i (x) = U b,i (x) = sin π (2i -1) x 2L ( 28
)
for i = 1, 2, 3...n ut or n u b . Substituting Eqn. ( 23)-( 25) in the kinetic and potential energy expressions and using Eqn. ( 13) for the expression of core longitudinal displacement lead to:

T = 1 2    ψ ξb ξt    T   M ww M T u b w M T utw M u b w M u b u b M T utu b M utw M utu b M utut      ψ ξb ξt    ( 29
)
and

V = 1 2    ψ ξ b ξ t    T   K ww 0 0 0 K u b u b 0 0 0 K utut      ψ ξ b ξ t    ( 30
)
The submatrices are presented in Annexe A. Lagrange's equations can be written as follows:

Equation of motion

d dt ∂L ∂ qk - ∂L ∂q k = Q k ( 31
)
where k = 1, 2, ..., N dof , L = T -V is the Lagrangian and Q k represents the generalized forces for the k th generalized coordinate, defined as:

Q k = δW δq k (32)
For an external force applied locally at x = x f , f (x, t) = f (t)δ(x -x f ), the expression of the virtual work after discretization is:

δW f = nw i δψ i (t) L 0 f (t)δ(x -x f )W i (x)dx (33)
Hence, from Eqn. (32), the generalized forces due to the transverse force are:

Q f =    f (t)W x=x f 0 0    (34)
For the core, the variation and discretization of Eqn. [START_REF] Cai | Vibration analysis of a beam with PCLD patch[END_REF] gives:

δγ = 1 h c U T t δξ t -U T b δξ b + d h c δψ ∂W ∂x (35)
which is introduced in Eqn. (21) and Eqn. (32) to obtain the generalized force: In the case of viscoelasticity, the expression of τ given by Eqn. ( 7) yields:

Q c =      -Bd L 0 τ W ,x dx B L 0 τ U b dx -B L 0 τ U t dx      (36)
Q c =      -Bd L 0 Gγ c + G ω γc W ,x dx B L 0 Gγ c + G ω γc U b dx -B L 0 Gγ c + G ω γc U t dx      (37)
While this expression simplifies when the core is linearly viscoelastic, i.e. G does not depend on γ, in the case of nonlinear viscoelasticity the expression has to be integrated at each step. Finally, the equation of motion for the generalized core behavior is:

  M ww M T u b w M T utw M u b w M u b u b M T utu b M utw M utu b M utut      ψ ξb ξt    +   K ww 0 0 0 K u b u b 0 0 0 K utut      ψ ξ b ξ t    =    Q c,w + Q f Q c,u b Q c,ut    (38)
5 Experimental study and discussion

Experimental set-up and specimen

The

Experimental results

A sinusoidal load of constant amplitude is applied to the moving mass of the shaker with stepped frequency. Different amplitudes are applied to observe the effect of the amplitude-dependency of the core properties on the overall beam behavior:

F 0 = [0.04 N, 0.1 N, 0.2 N, 0.4 N, 1 N, 2 N].
The resulting Frequency Response Function (FRF) is shown on Fig. 7(a) for the fourth accelerometer, at x = 238 mm. The same behavior is observed for the four accelerometers.

The FRF shows a strong dependency of the resonance peak on the load amplitude. Both the resonance frequency and peak amplitude decrease when the load amplitude increases. Moreover, the peak is asymmetrical.

This softening nonlinear behavior is consistent with the observation made in the first section that the shear modulus decreases with increasing shear strain amplitude.

The asymmetry of the peak prevents the use of a classical loss factor definition. However, the observed decreasing amplitude of the acceleration over load ratio indicates that the loss factor increases with increasing shear strain amplitude. This kind of softening core material can be interesting at higher load amplitudes as compared to classical core materials.

Comparison with the model

To compare the analytical model with the experimental data, Eqn. (38) is integrated in the time domain for different load amplitudes. The focus is made on the first bending mode. The number of modes used is n w = n b = n t = 2.

For fast computation, the spatial integration for the generalized force Q is made using the midpoint rule:

L 0 f (x)dx ≈ L n n k=1 f k + 1 2 n L (39)
The present computation is made with n = 4. Figure 7(b) shows the obtained FRF for the following storage modulus and loss factor functions:

G = 1.05 • 10 7 -1.5 • 10 10 | γ | (40) η = 1.2 • 10 -2 + 6.5 • 10 2 | γ | (41)
These results show that the proposed model allows to capture the behavior of a sandwich beam with core material nonlinearities. The modeling of G and η by a linear dependency with shear strain amplitude is sufficient to yield the observed behavior.

The parameters can be optimized to fit exactly the measures, but it was not the purpose of this work. The difference in the values between the shear test and the sandwich beam test can be explained by the heterogeneity of the material, that leads to a strong sensitivity to the sample size. The fact that very small strains are experienced inside the sandwich beam (from 10 -6 to 10 -4 ), smaller than what is measured in the material identification, can also be part of the explanation.

Conclusion

The vibration behavior of entangled cross-linked fibers used as sandwich core material was studied.

First, experimental shear tests showed that for this material the shear modulus decreases with the amplitude at low shear strains.

Lagrange's equations were used to derive an analytical model of a sandwich beam in which the core presents amplitude-dependent shear properties. The expression of strain energy of the core was replaced by a more general expression of the virtual work due to strain, allowing any material behavior to be taken into account.

A sandwich beam with entangled cross-linked fibers as core material was tested. The first mode was observed for increasing load amplitude. The resulting Frequency Response Function showed a softening nonlinear behavior.

This behavior could be reproduced with the proposed model. A linearly decreasing shear modulus and linearly increasing loss factor were used to obtain the Frequency Response Function, which is consistent with the shear test results.

The effect of different parameters of the material (density, type of fiber) need to be investigated to gain a better understanding of the phenomena leading to the observed softening behavior. The sensitivity to sample size will also have to be quantified to be included in the property models. From a more general point of view, the model introduced will allow the use of more complex models to describe the vibration behavior of the materials used in sandwich beams. These models include a more precise description of the shape of stress-strain loops when they are not elliptic.

M utw = h t -h b 8 ρ c h c B L 0 U t W T ,x dx (46)
M utu b = 1 4 ρ c h c B L 0 U t U T b dx (47)
K ww = (E b I b + E t I t ) L 0 W ,xx W T ,xx dx + k s W x=x f W T x=x f (48) K u b u b = E b h b B L 0 U b,x U T b,x dx (49)
K utut = E t h t B L 0 U t,x U T t,x dx (50)
The subscript , x denotes one derivative and , xx two derivatives with respect to x, respectively.

Figure 1 :

 1 Figure 1: Entangled cross-linked fiber material: shear sample.

Figure 2 :

 2 Figure 2: Experimental set-up for dynamic shear test.

Figure 3 :

 3 Figure 3: Evolution of shear modulus with the shear strain amplitude.

Figure 4 :Figure 5 :

 45 Figure 4: Clamped-free sandwich beam and lumped parameter electrodynamic shaker.

21) 4 . 2 . 4

 424 Virtual work due to the external load.

4. 3 . 1

 31 Lagrange's equations.

Figure 6 :

 6 Figure 6: Experimental set-up for flexural vibration.

  beam studied has a length L = 264 mm and width B = 40 mm. The sandwich is symmetrical with core thickness h c = 20 mm and facesheet thickness h t = h b = 2 mm. The core is made of entangled carbon fibers with a volume fraction f = 8.5%, cross-linked with epoxy resin for a total density ρ c = 180 kg/m 3 . The facesheets are made of seven plies of prepreg carbon fabric used in the aeronautical industry. The density and Young's modulus are measured experimentally: ρ b = ρ t = 1.5 • 10 3 kg/m 3 and E b = E t = 3.26 • 10 10 Pa.The beam is clamped at x = 0 and free at x = L. Excitation is applied with an electrodynamic shaker at x f = 37 mm. The excitation is close to the clamped end to minimize coupling between the shaker and the beam. Acceleration is measured at four points on the top of the beam which positions are x = [37 mm, 104 mm, 171 mm, 238 mm].The electrodynamic shaker's moving mass is m s = 2 • 10 -2 kg. The stiffness is k s = 576 N.

Figure 7 :

 7 Figure 7: Frequency response function of the sandwich beam with entangled carbon fiber core for different load amplitudes (a) experimental results (b) model for G = 1.05 • 10 7 -1.5 • 10 10 | γ | and η = 1.2 • 10 -2 + 6.5 • 10 2 | γ |.

Appendix A: Submatrices for the energy expressions


Samuel Hybois

Alice Siegel

Joseph Bascou

Nicolas Eydieux

Philippe Vaslin

El Ene Pillet

Pascale Fod

Christophe Sauret

Shoulder kinetics during start-up and propulsion with a manual wheelchair within the initial phase of uninstructed training

Keywords: Biomechanics, kinetics, rehabilitation, shoulder, wheelchair

Purpose: Wheelchair locomotion is constraining for the upper limbs and involves a set of motor tasks that need to be learnt by a novice user. To understand this integration process, we investigated the evolution of shoulder kinetics during start-up and propulsion within the initial phase of low-intensity uninstructed training. Materials and methods: Seventeen novice able-bodied subjects performed a 120-min uninstructed practice distributed over 4 weeks. During the initial and final sessions, upper limbs kinematics and hand-rim kinetics were continuously collected. Inverse kinematics and dynamics coupled to a three-dimensional linked-segment model were used to compute shoulder net moments. Results: Participants increased the speed of the wheelchair with practice. In average, an increase of shoulder net moments and mechanical work during the push phase was observed. Conversely, during the recovery phase, participants slightly increased shoulder power but maintained a similar level of shoulder loading. However, individual evolutions allowed the definition of two groups defined as: "increasers", who increased shoulder loading and mechanical work versus "decreasers", who managed to limit shoulder loading while improving the wheelchair speed. Conclusion: These findings underline that individual adaptation strategies are essential to take into account when designing a rehabilitation protocol for wheelchair users.

ä IMPLICATIONS FOR REHABILITATION

The learning process of manual wheelchair locomotion is essential for the assimilation of motor tasks leading individuals to select their propulsion technique. Novice users display different learning strategies: some people increase shoulder loading very early but others spontaneously manage to increase the wheelchair speed while maintaining a constant level of shoulder loading. Wheelchair rehabilitation programs should be individualized to take into account the subject-specific learning strategy.

Introduction

A manual wheelchair (MWC) is a medical device that allows people with motor disabilities to recover a part of their autonomy. Nowadays, MWCs are used by more than 600,000 persons in France, representing about 1% of the French population [START_REF] Ville | Les utilisateurs du fauteuil roulant en France. Le fauteuil roulant manuel: choix et r eglages [Manual wheelchair users in France[END_REF]. Higher rates were reported in some other countries, such as Great-Britain where MWC users represented about 2% of the population [START_REF]Improving services for wheelchair users and carers: Good Practice Guidelearning from the Wheelchair Services Collaborative[END_REF]. In Brazil, some authors reported about 5 million users from a population of about 200 million people, reaching a rate of about 2.5% [START_REF] Ackermann | Modeling and optimal control formulation for manual wheelchair locomotion: the influence of mass and slope on performance[END_REF]. However, propulsion through hand-rims fixed on the rear wheels, which is the widespread propulsion mode for MWC locomotion, is a highly demanding task for the upper extremity. As a consequence, it results in a high incidence of shoulder injuries in MWC users population [START_REF] Boninger | Shoulder magnetic resonance imaging abnormalities, wheelchair propulsion, and gender[END_REF][START_REF] Nichols | Wheelchair user's shoulder? Shoulder pain in patients with spinal cord lesions[END_REF][START_REF] Finley | Prevalence and identification of shoulder pathology in athletic and nonathletic wheelchair users with shoulder pain: a pilot study[END_REF][START_REF] Mercer | Shoulder joint kinetics and pathology in manual wheelchair users[END_REF]. The high mechanical loads and occurrence frequency of wheelchair-related tasks throughout the day are believed to be concomitant factors related to the apparition of musculoskeletal disorders [START_REF] Mercer | Shoulder joint kinetics and pathology in manual wheelchair users[END_REF][START_REF] Robertson | Pushrim forces and joint kinetics during wheelchair propulsion[END_REF][START_REF] Kulig | Shoulder joint kinetics during the push phase of wheelchair propulsion[END_REF][START_REF] Veeger | Load on the shoulder in low intensity wheelchair propulsion[END_REF][START_REF] Van Drongelen | Mechanical load on the upper extremity during wheelchair activities[END_REF]. However, stresses on the upper extremity are not only influenced by external loads such as the forces to overcome inertial forces and motion resistances [START_REF] Koontz | A kinetic analysis of manual wheelchair propulsion during start-up on select indoor and outdoor surfaces[END_REF], but also by the propulsion technique [START_REF] Boninger | Propulsion patterns and pushrim biomechanics in manual wheelchair propulsion[END_REF][START_REF] De Groot | Effect of Wheelchair Stroke Pattern on Mechanical Efficiency[END_REF][START_REF] Qi | Patterns of shoulder muscle coordination vary between wheelchair propulsion techniques[END_REF][START_REF] Slowik | The influence of wheelchair propulsion hand pattern on upper extremity muscle power and stress[END_REF] including both spatiotemporal parameters and joint configurations.

Hand-rim wheelchair locomotion is a set of motor tasks that needs to be learnt, often when being an adult. So to improve rehabilitation program efficiency, it is important to understand how these gross motor tasks are acquired. Numerous studies investigated the effect of various rehabilitation programs focusing on strength and physical conditioning [START_REF] Rodgers | Influence of training on biomechanics of wheelchair propulsion[END_REF][START_REF] Bougenot | Effects of a wheelchair ergometer training programme on spinal cord-injured persons[END_REF][START_REF] De Groot | Mechanical efficiency and propulsion technique after 7 weeks of lowintensity wheelchair training[END_REF][START_REF] Tordi | Effects of a primary rehabilitation programme on arterial vascular adaptations in an individual with paraplegia[END_REF] or on hand-rim wheelchair technique [START_REF] De Groot | Consequence of feedback-based learning of an effective hand rim wheelchair force production on mechanical efficiency[END_REF][START_REF] Leving | Effects of visual feedback-induced variability on motor learning of handrim wheelchair propulsion[END_REF][START_REF] Rice | Hand rim wheelchair propulsion training using biomechanical real-time visual feedback based on motor learning theory principles[END_REF]. Several studies investigated the effect of visual [START_REF] De Groot | Consequence of feedback-based learning of an effective hand rim wheelchair force production on mechanical efficiency[END_REF][START_REF] Leving | Effects of visual feedback-induced variability on motor learning of handrim wheelchair propulsion[END_REF][START_REF] Rice | Hand rim wheelchair propulsion training using biomechanical real-time visual feedback based on motor learning theory principles[END_REF] or haptic feedback [START_REF] Blouin | Characterization of the immediate effect of a training session on a manual wheelchair simulator with haptic biofeedback: towards more effective propulsion[END_REF] to reinforce the rehabilitation program efficiency with contrasted results. Others investigated the effect of uninstructed practice of hand-rim wheelchair propulsion [START_REF] Van Der Woude | Physical work capacity after 7 wk of wheelchair training: effect of intensity in able-bodied subjects[END_REF][START_REF] De Groot | Short-term adaptations in co-ordination during the initial phase of learning manual wheelchair propulsion[END_REF][START_REF] Roux | Shoulder movements during the initial phase of learning manual wheelchair propulsion in able-bodied subjects[END_REF][START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF] based on the assumption that subjects will naturally converge to an optimum technique minimizing energy expenditure for a given velocity. As a consequence, it may also be assumed that, at a self-selected comfortable velocity, shoulder loads would decrease throughout the learning process. This assumption is further supported by previous results where skilled hand-rim wheelchair users developed a lower propelling torque together with a higher mechanical power than unskilled subjects [START_REF] Hwang | Torque and power outputs on skilled and unskilled users during manual wheelchair propulsion[END_REF].

Generally, studies on wheelchair training program showed a decrease in oxygen consumption for a given external mechanical power, indicating an improvement in mechanical efficiency [START_REF] De Groot | Mechanical efficiency and propulsion technique after 7 weeks of lowintensity wheelchair training[END_REF][START_REF] Leving | Effects of visual feedback-induced variability on motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF]. Most of the studies focused on steady-state propulsion and investigated the changes in propulsion technique associated with the improvement in mechanical efficiency. In particular, an increase in cycle duration, contact angle and a change in force direction was observed [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF]. Other authors described different patterns of the hand during the propulsion cycle, which were correlated with the level of expertise: from arcing in novice to semi-circular in expert [START_REF] Boninger | Shoulder magnetic resonance imaging abnormalities, wheelchair propulsion, and gender[END_REF][START_REF] Slowik | The influence of wheelchair propulsion hand pattern on upper extremity muscle power and stress[END_REF]. In 2014, Vegter et al. [START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF] investigated the effect of a low-intensity uninstructed practice in naive ablebodied subjects and showed an increase in gross mechanical efficiency accompanied with a decrease of cycle frequency and negative work per push. However, in contrast with the results of Hwang et al. [START_REF] Hwang | Torque and power outputs on skilled and unskilled users during manual wheelchair propulsion[END_REF], Vegter et al. [START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF] found an increase in propelling torque within the training program, resulting in higher shoulder loads [START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF].

Wheelchair start-up is a highly constraining task for the upper extremity. Indeed, during start-up, shoulder loads were found to be between 1.8 and 3.5 times those observed during steady-state propulsion [START_REF] Koontz | A kinetic analysis of manual wheelchair propulsion during start-up on select indoor and outdoor surfaces[END_REF][START_REF] Westerhoff | Measurement of shoulder joint loads during wheelchair propulsion measured in vivo[END_REF]. In addition, start-up occurs from 200 to 350 times in a day [START_REF] Tolerico | Assessing mobility characteristics and activity levels of manual wheelchair users[END_REF][START_REF] Kloosterman | Clinical biomechanics exploration of shoulder load during hand-rim wheelchair start-up with and without power-assisted propulsion in experienced wheelchair users[END_REF]. As a consequence, it may be possible that during this task, subjects do not tend to exclusively decrease mechanical work but also tend to preserve their musculoskeletal integrity by limiting shoulder loads. In this case, the motor learning paradigm would be different between propulsion and startup. Indeed, some preliminary results have suggested that, over a certain level of shoulder loading, mechanical power decreased, whereas the mechanical work of the task still increased [START_REF] Sauret | Computation of the mechanical power of a manual wheelchair user in actual conditions: preliminary results[END_REF].

The purpose of this study was to investigate the evolution of shoulder kinetics during start-up and propulsion with a hand-rim wheelchair within the initial phase of a low-intensity uninstructed training. We hypothesized that net joint moments at the shoulder during start-up would quickly decrease in the course of the learning process. On the opposite, based on previous results [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF], an increase of shoulder net joint moment was expected for the propulsion task.

Materials and methods

Subjects

Following approval by the relevant ethics committee (CPP Paris VI Piti e Salp etri ere, France no. 2014-A01203-44), 17 able-bodied volunteers (4 females; 13 males) participated to this study. Subjects were 25.0 years old (SD: 3.4; range: 23-37 y.o.); height: 1.78 m (SD: 0.07 m, range: 1.63-1.97 m); weight: 71.6 kg (SD: 9.1 kg; range: 60-98 kg) and body mass index: 22.6 kg/m 2 (SD: 1.6 kg/m 2 , range: 20.0-25.4 kg/m 2 ). Able-bodied subjects were recruited because they provided a more homogeneous sample [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF] and because their adaptations would not be influenced by specific muscle weaknesses or lack of control.

Participants had no prior experience with MWC locomotion and none of them presented any upper limb pain or injury before and during the completion of the study. They were informed of the protocol and signed a written informed consent form prior to their participation, allowing both collection and use of their anonymous data reported in this manuscript.

Protocol

The design of the training program was largely inspired from the study of Vegter et al. [START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF] and consisted in about 120 min of uninstructed practice distributed over 4 weeks. The first and last 15 min of the training program were dedicated to measurement sessions.

The measurement sessions consisted in three blocks of about 5 min during which the participant had to perform a succession of tasks (including start-up, propulsion, slalom and U-turn with various turning radii) successively announced by an assistant. The volunteer was asked to perform these tasks at a self-selected comfortable pace. Two minutes of rest were awarded between blocks. No information was given to the participant on the way to perform these tasks. All the subjects used the same instrumented wheelchair during the measurement sessions.

Between measurement sessions, the participant was asked to perform twice a week, 16 min of practice split into 8 min blocks with at least 1.5 h between them. During these training sessions, subjects had to move freely on a hard smooth level ground, performing start-ups, propulsions, turns, slaloms and reverse propulsions. Training was performed using four hand-rim wheelchairs (Otto Bock Voyager; Invacare Kuschall KSL; Vermeiren D200 and Vermeiren Eclips 30 ) that differed in terms of geometry, wheel type, mass and inertia. The subject was asked to change of wheelchair every 4 min in order to limit accommodation to the settings of a single wheelchair and to favour the exploration of joint kinematics and kinetics through wheelchair characteristics. The subject received no information on how to improve his locomotion during the entire training program.

Finally, this training protocol was a low dose and low intensity uninstructed training program that should have prevented muscular and physiological adaptations. In other words, changes in biomechanical variables would be a motor learning only reflecting neurological adaptations.

Measurement procedure

Measurement procedure included the collection of both kinematic and kinetic data. Reflective markers were placed on both the subject and the wheelchair and their 3D locations were recorded at 100 Hz by 13 cameras of an optoelectronic motion capture system (Vicon system, hardware: T10 and T20S cameras; Nexus software; Oxford Metrics, Oxford, UK). Markers were placed on the spinous processes of the seventh cervical and eighth thoracic vertebrae, the manubrium, and the xiphoïd process for the torso; on the superior part of the acromion and on the medial and lateral epicondyles of the humerus for the arm; on the ulnar and radial styloïd processes for the forearm; and on the second and fifth metacarpal heads for the hand. Additionally, a technical platemounted markers cluster was attached to the arm and another technical marker was fixed on the distal third of the radius to assist in motion capture data collection. Wheelchair kinematics was recorded through four reflective markers placed on the chassis and on both rear-wheels centres.

Kinetic data were obtained through a wireless instrumented wheelchair allowing both forces and torques applied by the hand on the hand-rim to be recorded at a 100 Hz frequency (TSR-mesures, France [START_REF] Dabonneville | self-contained wireless wheelchair ergometer designed for biomechanical measures in real life conditions[END_REF][START_REF] Sauret | Effects of user's actions on rolling resistance and wheelchair stability during handrim wheelchair propulsion in the field[END_REF]). Signals were corrected from baseline values according to already detailed procedures [START_REF] Sauret | Zeroing of sixcomponent handrim dynamometer for biomechanical studies of manual wheelchair locomotion[END_REF][START_REF] Sauret | Dynamic calibration of a wheelchair six-component wheel dynamometer rolling on the floor[END_REF].

Synchronization between systems was ensured by knocking the instrumented hand-rim with a rubber hammer equipped with a reflective marker at the beginning of each block of locomotion. The shock created a synchronization event noticeable on both the hand-rim force signal and the hammer marker velocity.

Data processing

Start-up and propulsion cycles were identified in every block of recorded locomotion to provide kinematic and kinetic data for individual tasks.

Segment coordinate systems of the thorax, arm, forearm and hand were defined following the standards proposed by the International Society of Biomechanics [START_REF] Wu | ISB recommendation on definitions of joint coordinate systems of various joints for the reporting of human joint motion-Part II: shoulder, elbow, wrist and hand[END_REF]. Joint kinematics was obtained through a three-dimensional linked-segment model, using an inverse kinematics algorithm implemented in OpenSim [START_REF] Delp | OpenSim: opensource software to create and analyze dynamic simulations of movement[END_REF]. The model used included hand, forearm bones, humerus, clavicle, scapula and thorax. It was derived from Saul et al. [START_REF] Saul | Benchmarking of dynamic simulation predictions in two software platforms using an upper limb musculoskeletal model[END_REF], with the addition of 2 degrees of freedom (dof) between the clavicle and the thorax. These additional dof allowed to take into account the forward-backward and upward-downward motions of the glenohumeral joint centre with respect to the thorax reference frame.

Body segment inertial parameters were scaled from subjects' anthropometry preserving mass distribution of the generic model [START_REF] Saul | Benchmarking of dynamic simulation predictions in two software platforms using an upper limb musculoskeletal model[END_REF]. Then, shoulder net moments were obtained using an inverse dynamics process and were finally expressed in the thorax orthonormal coordinate system, centred on the humeral head centre, to allow clinical interpretation [START_REF] Mercer | Shoulder joint kinetics and pathology in manual wheelchair users[END_REF][START_REF] Kloosterman | Clinical biomechanics exploration of shoulder load during hand-rim wheelchair start-up with and without power-assisted propulsion in experienced wheelchair users[END_REF]. Definitions of parameters reported in the present paper are given in Table 1.

Data analysis

For each subject, all kinematic and kinetic data were split into single tasks, and time-normalized with respect to the task duration. Biomechanical variables (Table 1) were averaged over the different cycles (at least six per block) performed by each individual. Results were then averaged among the whole population.

Results

All participants followed the entire uninstructed learning protocol and performed the two measurement sessions. For clarity purposes, the results are presented for the initial (T1) and final (T2) blocks of the pre-learning test and for the final block (T3) of the post-learning session. In this way, the evolution between T1 and T2 represented the very short-term adaptation during the first 15 min, whereas the comparison with T3 gives the adaptation during the 3 weeks uninstructed training program.

Spatiotemporal parameters

Comparison of temporal parameters (cycle, push and recovery times) before and after the 3 weeks of uninstructed training program did not show major changes for both propulsion and startup tasks (Table 2). Indeed, for propulsion, cycle time remained constant between T1 and T3 and only decreased from 1.90 s at T1 to 1.80 s at T3 for start-up. However, in spite of this relative consistency in task duration, a slight increase of the recovery time accompanied by a shortening of the push time was observed in both propulsion and start-up tasks (Table 2).

Concerning MWC velocity, an increase in self-selected comfort velocity was observed, both during propulsion (T1: 0.70 m/s; T2: 0.86 m/s; T3: 1.02 m/s) and start-up (T1: 0.55 m/s; T2: 0.69 m/s; T3: 0.81 m/s).

Shoulder loading during the push phase

During the push phase, flexion (around the medio-lateral axis of the thorax) and internal rotation (around the longitudinal axis of the thorax) moments were the major components of the global shoulder net joint moment for both start-up and propulsion tasks. Peak values of these moments increased with learning for both propulsion and start-up tasks (Table 3). Peak value of the global net joint moment also increased with learning: from 15.4 Nm at T1 to 22.6 Nm at T3 for propulsion; and from 22.0 Nm at T1 to 30.8 Nm at T3 for start-up cycles. For both start-up and propulsion tasks, flexion angular impulse increased for short-term learning (between T1 and T2) but remained almost constant between T2 and T3 (Table 3). Finally, shoulder positive work during each push phase increased within the uninstructed training program for both propulsion (T1: 2.4 J; T2: 6.2 J; T3: 6.9 J) and start-up (T1: 5.6 J; T2: 9.1 J; T3: 12.2 J). A slight decrease of the shoulder negative work was observed, for both start-up and propulsion tasks.

Shoulder loading during the recovery phase

During the recovery phase, extension (around the medio-lateral axis of the thorax) and abduction (around the fore-aft axis of the thorax) moments were the major components of the shoulder net joint moments for both start-up and propulsion tasks. Peak moments and angular impulses followed the same trend for both tasks: after a slight increase within the first 15 min of practice (T1 vs T2), shoulder loads finally decreased at T3, below the initial values obtained in T1 (Table 4). Still, there was an overall increase in shoulder positive work during the recovery phase, even if it remained at the same level between T2 and T3.

Individual differences in learning strategies

Two main adaptation strategies, illustrated in Figure 1, were observed for the push phase during the learning process. Eleven subjects, classified as "increasers" (INC group), reached higher net shoulder moments at T3 than at T2, for both start-up and propulsion tasks. Individuals within the INC group increased the peak shoulder moment gradually for start-up (T2: þ34% with respect to T1; T3: þ81% with respect to T1) and for propulsion (T2: þ46%; T3: þ73%). The main contribution to this increase in shoulder moment was the flexion moment, with a shoulder push flexion impulse ranging from 10.7 Nm s (start-up, T1) to 19.3 Nm s (startup, T3). It resulted in an increase of shoulder positive work with learning among this group (Table 5).

The six other participants (DEC group) managed to limit their shoulder loading during the push phase, especially the flexion/ extension moment, while succeeding in increasing the MWC velocity. In average, the shoulder net moments within the DEC group were slightly modified between T1 and T2 for both propulsion (from 14.0 to 16.3 Nm) and start-up (from 25.9 to 25.8 Nm) and decreased at T3. The flexion moment largely contributed to this trend. For instance, the shoulder push impulse during start-up dropped to 9.7 Nm s at T3, with an initial value of 16.0 Nm s at T1. As shown in Table 5, shoulder positive work increased with a lower rate in the DEC group than in the INC group.

The recovery phase displayed a homogeneous trend for the whole population, since each subject reduced, or at least maintained, shoulder moments after the training period. It was not the case for the shoulder positive work during the recovery phase, which was higher in T3 than in T1 for 12 subjects, from both groups.

If average anthropometric characteristics showed that DEC individuals tended to be smaller and lighter (height: 1.76 m, SD: 0.04 m; weight: 66.8 kg, SD: 5.60 kg) than individuals of the INC group (height: 1.78 m, SD: 0.08 m; weight: 72.9 kg, SD: 8.10 kg), a significant overlap between the two groups was also observed.

Discussion

Spatiotemporal parameters

During this study, the task duration was not spontaneously modified by the users during the initial phase of the learning process. Propulsion cycles duration was in average higher than the values reported in the literature [START_REF] Leving | Effects of visual feedback-induced variability on motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF], which may be explained by the use of a field wheelchair in the present study, compared to studies performed on treadmills. For start-up, the duration of push phases was consistent with the study of Kloosterman et al. [START_REF] Kloosterman | Clinical biomechanics exploration of shoulder load during hand-rim wheelchair start-up with and without power-assisted propulsion in experienced wheelchair users[END_REF], who reported a push time of 1.3 s for experienced wheelchair users. However, during the very first minutes of practice (i.e., between T1 and T2), a slight increase of cycle time was observed, which was consistent with previous results on short-term learning [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF]. This highlighted the fact that the propulsion technique is not modified during the learning process following a continuous trend. Finally, while maintaining a similar total task time at T3, subjects spontaneously reduced the push time to increase the duration of the recovery phase. This natural evolution confirmed the acquisition of confidence and skills on the wheelchair with practice, revealing a more efficient handling of the MWC behaviour.

In contrast with cycle time, the self-selected comfort wheelchair velocity adopted by subjects increased with learning, for both short-term (T1 vs T2) and mid-term learning (T1 vs T3). This evolution was already mentioned during the very first minutes of practice in naive able-bodied subjects [START_REF] Roux | Shoulder movements during the initial phase of learning manual wheelchair propulsion in able-bodied subjects[END_REF]. However, many other studies investigating the effect of uninstructed MWC practice on propulsion technique were performed on a treadmill [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF], thus imposing a constant speed before and after the learning protocol. Given the conclusions of the present study, such protocols might have induced a bias in describing the natural evolution of over-ground MWC propulsion technique even if these studies remain interesting to describe the evolution at a controlled external mechanical power. In addition, if the treadmill velocity (1.11 m/s) chosen in previous studies [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF] could be suitable to experienced MWC users, our results (from 0.70 m/s at T1 to 1.02 m/s at T3) suggest that lower velocities should be preferred for naive young able-bodied subjects.

Shoulder loading during the push phase

At each step of the protocol, the push phase involved mainly flexion and internal rotation moments, for both start-up and propulsion, which confirms results of the literature for steady-state propulsion [START_REF] Mercer | Shoulder joint kinetics and pathology in manual wheelchair users[END_REF], with the same coordinate system. Our results also indicate that shoulder net moments were in average 1.4 times higher during start-up than during propulsion, which is slightly lower than the 1.6 to 3.1 ratio described by Koontz et al. [START_REF] Koontz | A kinetic analysis of manual wheelchair propulsion during start-up on select indoor and outdoor surfaces[END_REF] for hand-rim kinetics.

The increase of shoulder net moments and angular impulses within the first 15 min of practice, reported in the literature [START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF][START_REF] Vegter | Early motor learning changes in upper-limb dynamics and shoulder complex loading during handrim wheelchair propulsion[END_REF] for propulsion on a treadmill, was also observed in the present study. This increase not only occurred for propulsion and during the first minutes of practice, but also for start-up and throughout the 3 weeks of the uninstructed training period. Because the push time was not notably altered, it is mainly the increase of shoulder moments and work per push that enabled subjects to increase the linear speed of the wheelchair. Our hypothesis is thus confirmed for the propulsion task, but contrary to our expectations, start-up displayed a similar evolution with learning. This trend could be explained by the efforts developed by the user that may remain below the critical level of pain. This could lead to a learning process mainly driven by the building of self-confidence on the wheelchair rather than by a strategy favouring upper limbs preservation.

Shoulder loading during the recovery phase

Conversely to the push phase, the recovery phase has not often been investigated in previous biomechanical analyses of MWC locomotion, because most of these studies were focused on handrim kinetics [START_REF] Leving | Effects of visual feedback-induced variability on motor learning of handrim wheelchair propulsion[END_REF][START_REF] Van Der Woude | Physical work capacity after 7 wk of wheelchair training: effect of intensity in able-bodied subjects[END_REF][START_REF] De Groot | Short-term adaptations in co-ordination during the initial phase of learning manual wheelchair propulsion[END_REF][START_REF] Vegter | Inter-individual differences in the initial 80 minutes of motor learning of handrim wheelchair propulsion[END_REF][START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF] and there is no contact between the hands and the hand-rims during the recovery phase.

During the very first minutes of practice, the amplitude of net joint moments increased for the recovery phase. However, after the whole training protocol, subjects limited the articular loading in the backwards motion of their arms. The increase in shoulder positive work occurring in the meantime implied that the kinematics of the arm was the main factor influencing the motor learning in the recovery phase. After an initial session during which subjects made jerky movements for the transition between push and recovery phases, subjects chose a more continuous and economical motion between the end of the push and the next grasp of the hand-rim. It is consistent with the findings of other studies about the links between hand patterns and the level of expertise on the wheelchair [START_REF] Boninger | Propulsion patterns and pushrim biomechanics in manual wheelchair propulsion[END_REF][START_REF] Slowik | The influence of wheelchair propulsion hand pattern on upper extremity muscle power and stress[END_REF].

Individual differences in learning strategies

The emerging of INC and DEC trends was only noticeable after the 3-week period of practice. Indeed, for most subjects of the DEC group, the decrease in the shoulder loading did not occur at T2, but became obvious at T3. It can be explained by the fact that during the first 15 min of practice, subjects initially explored the range of motion before defining their personal propulsion strategy. That is why the learning process cannot be seen as continuous, especially between T1 and T2, but also between T1 and T3. However, it suggests that long-term learning strategies observed in studies about the very first minutes of MWC practice [START_REF] Vegter | Initial skill acquisition of handrim wheelchair propulsion: a new perspective[END_REF] may differ when considering a longer learning timescale. The learning paradigm that was hypothesized in this study was not confirmed, because individuals belonging to DEC or INC group were the same for start-up and propulsion, except for two subjects. Conversely, this study led to another conclusion considering our hypothesis: the motor learning process is not taskdependent (for start-up and propulsion) but subject-dependent. This implied that some subjects (INC group) imposed higher stress on their shoulder as they gain experience, while others (DEC group) used practice as a way to preserve their musculoskeletal integrity, regardless of the performed task.

That is why future research should deeper investigate the factors leading individuals to choose the INC or DEC learning strategy, especially by analysing muscle forces on the shoulder complex, for instance.

Limitations

There were some limitations in this study. First of all, because the protocol involved young able-bodied adults, application of the results to other populations should be made cautiously. Persons using a MWC temporarily after lower limb surgery, lower limb amputees or spinal cord injured patients with a lesion under the 12th thoracic vertebra can, however, be seen as populations for which the applications of the results are more straightforward. Conducting similar studies among a larger and more diverse population may help in broadening the learning spectrum.

Due to diverse practical reasons, only the right-side of the body was used for inverse dynamics. It might be interesting to investigate the possible impact of learning on the symmetry of upper limbs kinematics and kinetics. Because net joint moments are only global estimates of what is occurring at the joint level, future research should focus on muscular activations, in order to assess glenohumeral joint contact forces.

Conclusion

The results of this study show that biomechanical modifications of the propulsion technique occurred from the early stages of autonomous MWC practice. The push phase was critical, since the learning process enabled subjects to increase the wheelchair velocity, at the cost of higher shoulder moments and mechanical work. Contrary to our expectations, the evolution was similar for start-up and propulsion tasks. Conversely, skills acquisition enabled users to limit the shoulder loading during the recovery phase, through a better synchronization of the backward motion of the upper limb. Finally, two distinct trends, corresponding to different motor learning strategies, emerged: whereas some subjects increased shoulder net moments to reach a higher MWC velocity, others managed to limit shoulder loading. The motor learning paradigm seemed to be changing according to individuals, rather than to the task. These findings underline the need of individualizing rehabilitation programs, to help MWC users find their personal optimal propulsion technique. Future long-term longitudinal studies may also be conducted to evaluate the possible link between shoulder loading strategies and musculoskeletal disorders occurrence.

Figure 1 .

 1 Figure 1. Typical evolutions of flexion/extension moment patterns during the learning process, for the DEC (left) and INC (right) groups. Each graph represents the results for one individual from the population.

Table 1 .

 1 Biomechanical variables.

	Parameter	Unit	Description
	Push time	s	Duration of the push phase. Defined by the contact of the hands on the hand-rims
	Recovery time	s	Duration of the free-wheeling phase. Defined by the absence of contact with the
			hand-rims
	Task time	s	Total duration of the push and recovery phases
	Velocity	m/s	Velocity of the midpoint between rear wheel centre (mean value for propulsion cycles,
			final value for start-up cycles)
	Abduction/adduction shoulder moment M X	Nm	Net shoulder moment following the fore-aft axis of the thorax. Abduction moments are
			positive values
	Internal/external rotation shoulder moment M

Y Nm Net shoulder moment following the upward-downward axis of the thorax. Internal rotation moments are positive values Flexion/extension shoulder moment M Z Nm Net shoulder moment following the medio-lateral axis of the thorax. Flexion moments are positive values Global shoulder moment M G Nm Vectorial sum of M X , M X and M Z Shoulder push flexion impulse I þ (M Z ) Nm s Time integration of the shoulder flexion moment during the push phase Shoulder recovery extension impulse I À (M Z ) Nm s Time integration of the shoulder extension moment during the recovery phase Shoulder positive work W þ J Positive mechanical work of the shoulder moments Shoulder negative work W À J Negative mechanical work of the shoulder moments

Table 2 .

 2 Temporal parameters of MWC locomotion cycles.

		Propulsion	Start-up
		T1	T3	T1	T3
	Push time (s)	0.65 (0.10)	0.60 (0.05)	1.45 (0.23)	1.30 (0.15)
	Recovery time (s)	0.55 (0.13)	0.60 (0.18)	0.44 (0.12)	0.50 (0.19)
	Task time (s)	1.20 (0.20)	1.20 (0.21)	1.90 (0.31)	1.80 (0.28)
	Velocity (m/s)	0.70 (0.12)	1.02 (0.13)	0.55 (0.09)	0.81 (0.10)

T1: mean of initial pre-training trials; T3: mean of post-training trials. Standard deviations are presented between brackets.

Table 3 .

 3 Shoulder kinetic parameters during the push phase.

			Propulsion			Start-up
		T1	T2	T3	T1	T2	T3
	M max X M max Y M max Z M max G I þ (M Z ) (Nm s) (Nm) (Nm) (Nm) (Nm) W À (J) W þ (J)	1.5 (2.0) 3.1 (1.7) 9.4 (3.6) 15.4 (3.4) 2.2 (1.2) 2.4 (1.6) 2.4 (1.0)	5.2 (6.5) 4.2 (1.8) 17.0 (8.1) 20.9 (9.2) 4.6 (2.9) 2.1 (1.2) 6.2 (5.1)	4.6 (5.8) 5.5 (2.5) 18.7 (10.5) 22.6 (10.1) 4.5 (2.7) 1.9 (0.9) 6.9 (4.6)	3.6 (3.8) 5.9 (2.3) 19.3 (8.1) 22.0 (7.6) 12.5 (6.8) 1.9 (1.7) 5.6 (2.5)	5.0 (4.1) 8.8 (2.5) 23.6 (8.0) 26.3 (8.2) 15.3 (6.3) 1.6 (1.3) 9.1 (4.5)	5.0 (4.4) 11.3 (2.6) 28.1 (9.7) 38.0 (9.5) 15.9 (6.0) 1.6 (1.3) 12.2 (5.2)
	T1: mean of initial pre-training trials; T2: mean of final pre-training trials; T3: mean of post-training trials. Standard devi-
	ations are presented between brackets.

Table 4 .

 4 Shoulder kinetic parameters during the recovery phase.

			Propulsion			Start-up
		T1	T2	T3	T1	T2	T3
	M min X M min Y M min Z M max G I À (M Z ) (Nm s) (Nm) (Nm) (Nm) (Nm) W À (J) W þ (J)	À5.4 (2.5) À1.5 (1.0) À12.3 (3.6) 13.5 (3.7) À5.1 (2.0) À0.2 (0.2) 6.3 (2.3)	À5.9 (1.9) À2.0 (1.0) À13.5 (4.0) 14.6 (4.0) À6.4 (2.5) À0.6 (0.5) 7.6 (2.4)	À5.5 (2.3) À3.2 (1.8) À12.0 (3.9) 13.4 (4.2) À4.5 (2.0) À0.2 (0.2) 7.7 (2.6)	À5.5 (3.3) À1.2 (0.8) À11.6 (3.8) 12.9 (4.3) À4.1 (1.9) À0.1 (0.1) 5.8 (2.6)	À6.3 (2.2) À1.7 (1.0) À12.6 (3.5) 14.1 (3.4) À5.5 (2.4) À0.3 (0.2) 7.4 (2.5)	À5.3 (2.4) À2.2 (0.9) À10.5 (3.6) 11.9 (3.7) À3.7 (2.2) À0.2 (0.3) 7.2 (3.0)

T1: mean of initial pre-training trials; T2: mean of final pre-training trials; T3: mean of post-training trials. Standard deviations are presented between brackets.

Table 5 .

 5 INC/DEC kinetic parameters during the push phase.

			INC group			DEC group
		T1	T2	T3	T1	T2	T3
	Start-up: M max G Propulsion: M max (Nm) G (Nm) Start-up: I þ (M Z ) (Nm s)	19.9 (7.0) 16.2 (3.6) 10.7 (6.6)	26.6 (8.6) 23.7 (10.2) 15.4 (6.8)	36.1 (6.6) 28.0 (8.3) 19.3 (3.6)	25.9 (7.6) 14.0 (2.7) 16.0 (6.0)	25.8 (8.2) 16.3 (5.0) 15.1 (5.9)	21.1 (4.9) 12.8 (3.6) 9.7 (4.1)
	Propulsion: I

þ (M Z ) (Nm s) 2.2 (0.7) 5.0 (3.2) 5.7 (2.2) 2.4 (1.8) 4.0 (2.4) 1.1 (1.7) Start-up: W þ (J) 5.0 (2.4) 9.2 (5.0) 14.6 (4.7) 6.6 (2.6) 8.8 (4.1) 7.7 (2.1) Propulsion: W þ (J) 2.5 (1.0) 7.5 (6.0) 9.1 (4.3) 2.2 (1.1) 4.1 (1.8) 2.9 (1.1) T1: mean of initial pre-training trials; T2: mean of final pre-training trials; T3: mean of post-training trials. Standard deviations are presented between brackets.

Disclosure statement

The authors report no declaration of interest.