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. Nous-utilisons-la-matrice, 4) pour réécrire I'expression (A.24) sous Ia forme suivante: {ôq}' IQJ'tcr No s NolleJ {q"} + {ôq}' Je.l'({rry} + {rxr, p.5

. Où-la-matrice-de-rigidité-globale, ainsi que les seconds membres {Ff;} contiennent les contributions dues au contact: K'l : Ix!I+ Kfl tFit):{Fite}+{F#"} {F#"} tKfl A.2, Termes du contact dans Ie cas où Ia normale varie A.2

. Nous-la-réécrivons-de-la-ma, nière suivante: n-I

. {aq, }t{Ê-} : 1ôq"}t[rNg]{.4} + {ôq

. Et-donc-le-problème-À-i-'ordre-n, L6) se réécrit: 1xfe11q"1, {F} + {Firs}+ Ia.l'{rru} {r#c} (A.31) (4.32) (A.33

A. Annexe, Phase de prêdiction c(p){q(p)}'{"'.(p)} -ry

. Ori-{arf, forme discrétisée de I'incrément des efforts de contact qui est explicitée dans le chapitre (4) et {Aq"} est I'incrément des déplacements à I'ordre n

. Si-on-utilise-i-'algorithme-de-newton-ra, phson d'ordre étevé (4.18) , p{n1est Ia matrice de rigidité tangente classique définie à I'itération actuelle c'est à dire évaluée au point (Ur,Àr): l#n): i" / rt""t'rotFol+ tcl'tspltcl) do ":rd

. Si-c-'est-i-'algorithme-de-newton, Raphson Modifié d'ordre élevé qui est utilisé, ators [ffe1 désigne la matrice de rigidité tangente calculée à la première itération et {AFfle} s'écrit: {argrel : -Ï" / (t"t'ttos,} + taSfzlt1 + [Bp],({as#} + lasfrz)ly) ao J:ld a' '-' ' \ '3 )' ' I r

. Ath@, Lu^(ù.e@) + a'ny1o! A, pp.1-6

L. Relation, 9) se réécrit donc: ] aR-(e) : r(p)[aÂ, (p)r(p) + tr@ @) TIOWrA + wrntçra4@) *i oo-r(e)aq(e)] i:l (8.13), pp.20-28

. Anl, ,(pr) : tI ot-r tr ) ari'(pr )

{. Soit and . Le-vecteur-défini-par, {arNr} : {rR"(p)K#} + {rr(P)aN"I} + {faRJ\

. Nous-utilisons-la-matrice, 4) pour réécrire l'expression (8.23) sous Ia forme suivante: {aôq}' IQJ'tcr Ne s r.TPltQ"l {aq"} + {aôq}t {g:l'(t.4lË'}ll4lx!})

. Nous-la-réécrivons-de-la-manière-suivante, AR-(p) : an"(l'xr |to(p)) a aetr)@)f @) k*@\"o(ù + rerrtç)f (1D) a'#(p) n, p.1

. Avec, 16), nous {AÈ"} : {(r'P)(er)}

@. Et-donc-le-problème-À-i-'ordre-n, 20) se réécrit: 1t'011aq,) -[e.]'[rDrp]{an"} . : aÀ,"{F} + {aF#s}+ Je, pp.28-36

B. Annexe, Phase de corraetion Nous reprenons à présent l'expression de {ARr} en chaque naeud de contact e @.L2) Nous la réécrivons sous la forme suivante: c tù{* tù}t{4u, AR*(p) : -A4 @) (8.33)