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, Chapitre 3 SLrr la défot-mation d'ttn frl éIastique pat' trois clisclrre^s... 1. Introduction Dans ce qui suit, on propose d'étudier la position d'équilibre de trois disques identiques rigides (Dt.,I < i < 3) roulant sur un fil élastique

, on se place dans un repère orthonormal dont I'axe des ordonnées est dirigé vers le bas. On représente par I'intervalle 0: (0, 1) la position initiale du fil éiastique, Comme dans le cas de deux disques identiques

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*. R\at, , 2001.

, Comparison between the two ca,ses (i) and (ii)

, Appendices 130

A. A. , Interchanging the order of integration Let f ,g be continous functions respectively on IR+ x R+, R+. Then we have: Ir' Ir" f (u,s)s(s)itsilu = Io' {1,' ï{u,s)d,u}s(s)d,s

. Fu, We can see that the integral, s)g(s)d,sd,uis taken over the triangle in Jo Jo figure al (o1) left-hand side of (al) gure a1

, Therefore, vr'e can easily deduce the formula (al). l right-hand sicle of (al)

. Gronwall, be continous and C bea nonnegative number If 7t f(ù<C+ | s(s)/(s)ds, 0(t<a, Jo then Proof' f t Suppose first that C > 0. Divide by C + Jo s@f?)ds and multiply by s(t) to obtain: 7t f (t)g(t)l{c * Jo gG)f G)ds} < g(t)

, An integration from 0 to t yields: 1tfi tog{(C+ / e(s)/(s)ds)lC}< | s(s)ds

, Jo Jo or ï(t) <c * [' gG)ïG)d,s 3C"xp{ / s(s)ds}

J. Jo,

, take the limit as C + O throught positive values. This completes the proof. 1.t T(t) < Cexp{ | s(s)ds), pp.0-1

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