Vibrations of long repetitive structures by a double scale asymptotic method

In this paper, an asymptotic two-scale method is developed for solving vibr ation problem of long periodic structures. Such eigenmodes appear as a slow modulations of a periodic one. For those, the present method splits the vi bration problem into two small problems at each order. The first one is a p eriodic problem and is posed on a few basic cells. The second is an amplitu de equation to be satisfied by the envelope of the eigenmode. In this way, one can avoid the discretisation of the whole structure. Applying the Floqu et method, the boundary conditions of the global problem are determined for any order of the asymptotic expansions.